Reliability-based Design Methods Research Articles

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.

Time- and Space-Dependent Reliability-Based Design With Envelope Method

AbstractDeterministic optimization may lead to unreliable design results if significant uncertainty exists. Including reliability constraints in reliability-based design (RBD) can solve such a problem. It is difficult to use current RBD methods to deal with time- and space-dependent reliability when responses vary randomly with respect to time and space. This study employs an envelope method for time- and space-dependent reliability for the optimal design. To achieve high accuracy, we propose an inverse envelope method that converts a time- and space-dependent limit-state function into a time- and space-independent counterpart and then use the second-order saddlepoint approximation to compute the probability of failure. The strategy is to find an equivalent most probable point for a given permitted probability of failure for each reliability constraint. To achieve high efficiency, we use a sequential optimization process to decouple the double-loop structure of RBD. The overall optimization is performed with a sequence of cycles consisting of deterministic optimization and reliability analysis. The constraints of the deterministic optimization are formulated using the equivalent most probable points. The accuracy and efficiency are demonstrated with four examples, including one mathematical problem and three engineering problems.

Durability of FRP bars and FRP bar reinforced seawater sea sand concrete structures in marine environments

• Combination of the alkaline and marine environments aggravates the degradation of mechanical properties of FRP bars. • The variability of mechanical properties of FRP bars shows an increasing trend. • Bearing capacities of FRP bar reinforced SSC components decrease and the failure mode changes. • Environmental reduction factors in current codes are conservative. Fiber reinforced polymer (FRP) is one of the alternatives to steel, and can be directly used in seawater sea sand concrete (SSC). But FRP material still has durability problems in corrosive environments, which is one of the hot and difficult issues of current research. Based on the existing literature, this paper combs the degradation of the long-term property of FRP bars and the durability of FRP bar reinforced SSC (FRP-SSC) structures in the marine environment. The combination of the alkaline environment of concrete and the salt ions in sea sand and seawater aggravates the dissolution of the resin and debonding of the fiber-resin interface of FRP bars (particularly for BFRP bars). Therefore, the mechanical properties of FRP bars degrade with high variability which leads to the reduction of bonding between SSC and FRP bars. Additionally, the bearing capacity of FRP bar/steel-FRP composite bar (SFCB) reinforced SSC components in marine environments decreases, and the failure mode of components changes. The current codes use environmental reduction factors in the durability design of FRP bar reinforced concrete structures. But the values of reduction factors are conservative. Finally, it is pointed out that the quantification of variability for the mechanical property of FRP bars after long-term service, and the time-dependent reliability-based design methods of components are needed for further studies.

Additional insights to EC7 from the application of reliability-based design methods: the case of debris flow protection structures

Debris flows are dangerous natural processes that cause extensive damages to infrastructures and urbanized areas and can lead to loss of human lives. Their unpredictability, their extremely high motion and their magnitude are the main causes of these harms. Mitigation measures are fundamental for reducing the associated risk and protecting infrastructures in mountainous areas. Their design is still an open issue: there are many formulations to evaluating impact pressure. Moreover, the uncertainties in the determination of flow characteristics (velocity and thickness) are significantly high and difficult to quantify. In the European Union, the design of any type of structures involved in rock mechanics field must comply with EN-1997 Geotechnical Design (CEN 2004) (EC7). For debris flow countermeasures, EC7 requirements are very difficult to apply in practice since partial safety factors are not provided for these phenomena. However, the basic philosophy of reliability-based design (RBD), as defined in EN1990 (CEN 2002) may be a suitable and complementary approach to provide geotechnical structures with a uniform probability of failure. Reliability Based Design (RBD) can provide additional insights to EC7 design and can be applied when partial factors have still to be proposed (by EC7) to cover uncertainties of less common parameters, as in case of debris flow countermeasures. This paper presents an analysis of the advantages and limitations on the applicability of RBD approach to debris flow countermeasures, by using the first-order reliability method (FORM). In particular, data availability, the possibilities for analysing data in a statistical framework and the choice of performance function are the main limitation of the method, which force to make assumptions regarding statistical distribution of the considered parameters. A sensitivity analyses, comparing different equations, commonly used for debris flow impact pressure estimation, were performed for quantifying the effect of the selected performance function on the RBD results.

Reliability based design of shaft for gearbox

In this article, the author presents research on the reliability-based design methods of mechanical systems, includes 3 stages: the first, the fault tree analysis (FTA) method is utilized for reliability analysis of single and double reducer into the logic diagram; the second that apply the probabilistic design methods, including moment matching method (MMM), first-order reliability method (FORM), and Monte Carlo simulation (MCS) for reliability-based design and to analyze the components of the mechanical transmission systems, and the last sensitivity analysis is performed in the post-design stage after a design solution is identified. The study also aims to set up RADME (Reliability-based analysis and design of the mechanical systems) computer programs applying the aforementioned methods. The reliability design problem was surveyed as reliability-based design optimization with one design variable.

Reliability Analysis of Bored-pile Wall Stability Considering Parameter Uncertainties

In geotechnical engineering, bored-pile wall stability is evaluated using deterministic design methods based on safety factors to establish a margin against failure. In recent years, reliability-based design methods have been adopted to include uncertainty in the assessment of bored-pile wall stability as well as in the calculation of the feasible embedment depth of the walls. In this study, an expanded reliability-based design approach, along with finite element analysis, was applied to conduct parametric analyses of bored-pile wall stability. In serviceability limit state design framework, the results indicate that cohesion and groundwater level are factors that significantly affect bored-pile wall stability. Moreover, high variability in the cohesion range causes great uncertainty to determine the embedment depth of bored-pile wall. The feasible embedment depth can reach 4 times the free height considering the maximum coefficient of variation (50 %) of the cohesion. In turn, when the groundwater level is located at the retained ground surface, the horizontal displacement of the upper end of the wall reaches 15.2 mm, i.e., 0.0038 times the free height of the wall, for which the soil mobilizes active earth pressures. It was also found that the resolution of probabilistic results is highly influenced by the number of iterations in Monte Carlo simulations.

Decoupled reliability-based geotechnical design of deep excavations of soil with spatial variability

This paper presents a general decoupled method for reliability-based geotechnical design that takes into account the spatial variability of soil properties. In this method, reliability analyses that require a lot of computational resources are decoupled from the optimization procedure by approximating the failure probability function globally. Failure samples are iteratively generated over the entire design space so that their global distribution information can be extracted to construct the failure probability function. The method is computationally efficient, is flexible to implement, and is well suited for geotechnical problems that may involve sophisticated models. A design example of two-dimensional deep excavation against basal heave is discussed for Singapore marine clay where the density and normalized undrained shear strength of soil mass are modeled as random fields. Results demonstrate that the proposed method works well in practice and is advantageous over the coupled or locally decoupled reliability-based geotechnical design methods.

Effects of bottleneck blockage on the resilience of an urban stormwater drainage system

ABSTRACTSevere urban floods have frequently caused social and financial infrastructure problems due to a lack of hydraulic capacity of urban drainage systems or the structural failure of network elements. Conventional, reliability-based design methods of drainage networks only provide acceptable performance under expected conditions of loading. In this study, a new approach based on resilience is proposed for the rehabilitation of urban drainage systems; this approach is able to maintain a predefined service level under both external and unexpected internal loads, including different scenarios of blockage and a severe rainfall event. The performance of different design methods: resilient, non-resilient, deterministic and uncertain models, is comprehensively compared and the most appropriate approaches are introduced. The results show that the new resilient approach outperforms conventional design/rehabilitation methods in terms of flood volume reduction. Additionally, the single blockage scheme and the Poisson distribution-based model show the best performance to enhance the resilience of the network.

Full probabilistic geotechnical design under various design scenarios using direct Monte Carlo simulation and sample reweighting

It is common to encounter a diversity of design scenarios with different values and statistics of loads and/or geotechnical parameters in design practice of engineering geology and geotechnical engineering. Maintaining a uniform level of reliability close to the target reliability under various design scenarios is a key goal of geotechnical reliability-based design (RBD), which is a fundamental challenge for semi-probabilistic RBD methods because a wide range of design scenarios can be involved in geotechnical design practice. This paper develops an efficient RBD updating approach under a full probabilistic design framework, which combines direct Monte Carlo simulation (MCS) with a sample reweighting technique to efficiently obtain final designs under different design scenarios by a single simulation run, avoiding repeatedly performing direct MCS for different design scenarios. This leads to considerable computational saving, particularly as the number of design scenarios concerned is large. The proposed approach is illustrated through a pad foundation design example and a rock slope design example. Results show that the proposed approach properly calculates failure probabilities of different possible designs under different design scenarios by a single run of direct MCS, provided that sufficient failure samples are generated in the direct MCS run to cover failure regions of different design scenarios, which are used to determine the updated final designs. Reliability of the final designs obtained from the proposed approach for various design scenarios is generally uniform and is close to the target reliability level prescribed for the design purpose. Moreover, the proposed approach is able to link site investigation efforts to design saving in an efficient and straightforward manner. It also provides insights into the sensitivity of the final design to uncertain parameters involved in design.

Robust reliability-based optimization with a moment method for hydraulic pump sealing design

Reliability-based design (RBD) is widely applied to meet the reliability requirement at a reduced cost. Current RBD methods require complete distributions of basic random variables, but in real applications, such as the sealing design for hydraulic piston pumps, only statistical moments may be available. The reliability obtained from the RBD may also be sensitive to possible variations in distribution parameters or moments of basic random variables. To deal with these two challenges in the sealing design for hydraulic piston pumps, this study employs a robust reliability-based design methodology based on statistical moments. The optimization not only minimizes the sealing area of friction pairs, but also the sensitivity of the reliability. As a result, the reliability requirement is maintained with the increased robustness of the reliability. An improved higher moment (up to the forth moment) is also employed for higher accuracy with the associated analytical sensitivity indexes. The robust RBD methodology is successfully applied to the design of the seal of the slipper-swash plate friction pairs for a hydraulic piston pump. The application includes the reliability analysis, reliability sensitivity analysis, and robust RBD. The results demonstrate that robust RBD can produce more robust reliability than a traditional RBD methodology.

Reliability-based design of stiffened plates in ship structures subject to wheel patch loading

Reliability-based design methods are among the engineering tools for design of structures considering uncertainties in the design variables such as loads, material properties, manufacturing tolerances, etc. In design of stiffened plates of ship structures, there are some sources for uncertainties regarding the loads due to the highly nonlinear environment of the ocean. In addition to the environmental uncertainties, internal forces decrease the reliability of design due to the vagueness of items, which are included in the deadweight. Cargos with patch loading pattern like wheel and steel coil are common type of loading in some ships. These cargos are usually transferred by ferries, general cargo ships and bulk carriers. The application of reliability method in design of such structures against this local loading pattern has not been studied so far. Therefore, in this paper, the stochastic method and plastic formulas have been explained in designing steel panels subject to the patch loading. As a case study, stiffened plate structure of the ships subject to the truck wheel was investigated considering all design parameters as random variables. Bound theorem model for the patch loads is used to calculate the plastic load carrying capacity of the plate. First Order Second Moment (FOSM), First Order Reliability Method (FORM) and Importance Sampling method have been employed in the analyses. Both probability of failure and reliability index are calculated for different axel load distribution functions. The total failure probability was determined based on the rule of total probability. Sensitivity analyses for all parameters were carried out and the effects of coefficient of the variation of dominant variables were studied.

Role of redundancy in simplified geotechnical reliability-based design – A quantile value method perspective

In this study, we show that existing simplified reliability-based design methods, including the partial factor method and the quantile-based method (quantile value method; QVM), are not robust over design scenarios with variable degree of redundancy. This variable degree of redundancy is quite common in geotechnical design. For instance, a pile group has more redundancy than a single pile. Less obviously, a pile embedded in multiple soil layers has more redundancy than one embedded in a single soil layer. Implementing a fixed set of partial factors or a fixed quantile will not produce designs with uniform reliability indices. This paper shows that the degree of redundancy can be effectively quantified by the concept of an “effective random dimension” (ERD), and it proposes a practical method of estimating ERD for reliability-based design. It is shown by numerical examples that by incorporating ERD, the ERD-QVM outperforms the original QVM in achieving a more uniform reliability level across different design scenarios.

Stochastic assessment for the behaviour of systems of dry soil mix columns

The mechanical properties of dry soil mix (DSM) columns can be highly variable. Variability can be accounted for in the construction specification for deterministic design and directly in reliability based design. Design methods and specifications to date adopt simplifications that do not take the variability of the columns fully into account. This paper uses both simple and advanced probabilistic methods to assess the performance/failure and system redundancy of dry soil mix columns. Reliability-based design methods and examples are given for the design of column strength and the adjustment of the column spacing to achieve a target probability of unacceptable performance or failure. An acceptance criteria chart is developed. The pull-out resistance tests on the DSM columns constructed for the Ballina Bypass motorway construction project in NSW Australia are compared to the chart to provide guidance with respect to acceptance criteria required to achieve the desired performance.

Reliable Support Design for Excavations in Brittle Rock Using a Global Response Surface Method

Spalling damage can pose significant risks during the construction of underground excavations in brittle rock. While deterministic analyses have traditionally been used in the design of these structures, reliability-based design (RBD) methods provide a more rational approach to quantify spalling risk by directly incorporating input uncertainty into the design process and quantifying variable ground response. This paper presents a new RBD approach to evaluate the excavation response and support performance for a tunnel in brittle ground. Guidance for the selection of appropriate parameters for variable brittle materials is provided using a combination of the damage initiation and spalling limit method and theories of microcrack initiation. System performance is then evaluated using a proposed global response surface method (GRSM) coupled with the first-order reliability method, random sampling and finite element analysis. The proposed GRSM provides a computationally efficient way to evaluate the probability of failure for various limit states, allowing for the selection of appropriate design parameters such as minimum bolt length and required bolt capacity during early stages of design. To demonstrate the usefulness of this approach, a preliminary design option for a proposed deep geologic repository located in Canada was assessed. Numerical analyses were completed using finite element modeling to determine the depth of spalling around the excavation and support loads over the range of possible rock mass and in situ stress conditions. The results of these analyses were then used to assess support performance and make support recommendations.

Linking Site Investigation Efforts to Final Design Savings with Simplified Reliability-Based Design Methods

AbstractThis study aims to address how efforts spent in geotechnical site investigation can be linked in an objective and defensible way to the final design dimensions of a geotechnical structure. A previously developed pad foundation example is adopted for illustration. Three aspects of geotechnical information are addressed: a variety of in situ and laboratory tests, precision of the tests, and local experiences in the estimation of soil properties that permit reduction of transformation uncertainties between test indexes and design parameters. Four simplified reliability-based design methods based on partial factors and quantiles are calibrated for the pad foundation. Their performances in linking site investigation efforts to final design savings are studied. Among these methods, the quantile methods are found to be the most responsive to the availability of geotechnical information, whether in the form of site investigation efforts or local experiences. The quantile methods can fill a critical gap in...

Design Methodology for Axially Loaded Auger Cast-in-Place and Drilled Displacement Piles

AbstractWith the increasing use of augered cast-in-place and drilled displacement piles in new construction, it is important that proper design parameters be incorporated when evaluating pile performance using reliability-based design methods. Although augered piles can be distinguished from bored piles, including drilled shafts, and driven piles by the magnitude of the effective stress changes they produce in the vicinity of the pile during construction, the current design methods for augered piles generally use the same design methods as those used for bored or driven piles. To enhance the efficiency of the augered piles, a unique design method must be developed. This paper focuses on developing a design methodology for axially loaded augered piles installed in predominately sandy soils using the t-z method. To develop the design parameters for augered piles, back-calculation of the t-z parameters was conducted using static load–test data. The data from 17 static pile load tests conducted on augered pil...

Reliability-Based Design of Augered Cast-in-Place Piles in Granular Soils

AbstractAlthough a variety of methods exist to estimate axial capacity for augered cast-in-place piles, they are generally limited to allowable stress design (ASD) procedures, with little consideration of design reliability. This paper describes the addition of static loading test results to a global augered cast-in-place pile data set to assess the accuracy of new and existing design methods and to address the current lack of reliability-based design methods for augered cast-in-place piles. The new static loading tests in western Washington were carried out on piles installed in granular materials, with pile diameters and lengths ranging from 0.41 to 0.51 m and 9.5 to 29 m, respectively. The preparation of beta coefficients and unit toe bearing resistance values is discussed within the framework of strain-dependent composite tangent moduli and observed residual loads. New relationships for the beta coefficient and toe bearing resistance values are proposed, and the accuracy of new and existing design pro...

Reliability-Based Optimization and Robust Design of a Coil Tube-Spring with Non-Normal Distribution Parameters

This paper proposes the application of reliability-based optimization and robust design methods of a coil tube-spring. The perturbation method, the Edgeworth series, the reliability-based optimization, the reliability sensitivity technique, and the robust design method are employed to present practical and effective approaches of reliability-based optimization and robust design for coil tube-spring with non-normal distribution parameters, on the condition of the known first four moments of the original random variables. Theoretical formulae for reliability-based optimization and robust design are obtained. The respective programmes can be used to obtain the reliability-based optimization and robust design parameters of a coil tube-spring with non-normal distribution parameters accurately and quickly.

Probabilistic engineering analysis using the NESSUS software

The development of reliability-based design methods requires the use of general-purpose engineering analysis tools that predict the uncertainty in a response due to uncertainties in the model formulation and input parameters. Barriers that have prevented the full acceptance of probabilistic analysis methods in the engineering design community include availability of tools, ease of use, robust and accurate probabilistic analysis methods, and the ability to perform probabilistic analyses for large-scale problems. The goal of the reported work has been to develop a software tool that fully addresses these three aspects (availability, robustness and efficiency) to enable the designer to efficiently and accurately account for uncertainties as they might affect structural reliability and risk assessment. The paper discusses the NESSUS probabilistic engineering analysis software with specific sections on the reliability modeling and analysis process in NESSUS, the robust and accurate solution strategies incorporated in the available probabilistic analysis methods, and several application examples to demonstrate the applicability of probabilistic analysis to large-scale engineering problems.

Influence of proof loading on the reliability of members

Proof loading concept is a recognized quality control technique to improve the characteristics of the lower tail of strength distributions of structural timber products. Very few comprehensive studies are available to quantify the effectiveness on the use of proof loading relating to the choice of proof load level, the potential damage on the members resulting from the application of proof load, and the improvement of performance in the context of reliability based design methods. One of the difficulties is the need of a rather large sample size for an experimental-based study to develop statistically meaningful solutions. This paper illustrates the use of damage accumulation model and reliability based design analyses to quantify the effectiveness of proof loading. The performance of No. 2 and better Western Hemlock 38×140 mm dimension lumber in bending is considered. Damage accumulation laws are established to consider the residual strength of members that survived a proof load. Reliability analyses are conducted to compare the performance of proof-loaded and non-proof loaded members subject to snow load conditions in two locations in Canada. For a given reliability index (β), the improvement of performance can be quantified as characteristic strength adjustment factors for proof loading in terms of the ratio of the performance factors (φ) between the original and proof load material. Conversely the gain in reliability, β, for a given φ value is also apparent. The adjustment factors depend on the proof load level, the β level, and the distribution selected for fitting the strength data after proof loading.