Failure Probability Level Research Articles

A Bayesian-Based Credible and Reliable Control Method for Active Vibration Suppression of Structures with Uncertain Parameters

As a classical design method for addressing uncertainty in control systems, robust control is designed with the goal of complete reliability of the control system. Nevertheless, the pursuit of robustness can inadvertently give rise to considerable performance degradation, rendering the resultant closed-loop control system excessively conservative in its design. Reliability-based design methods serve as an effective approach for managing uncertainties, which will achieve a balance between the robustness and performance of the active control system by taking advantage of the characteristic of reliability that allows for a certain level of permissible failure probability. The accurate quantification of uncertainty is the key to achieving reliable control, but common uncertainty quantification methods cannot achieve credible quantification of uncertainty in small samples and update the distribution of new samples. In this paper, a nonprobabilistic Bayesian inference based interval uncertainty quantification method is proposed. This method updates the distribution of interval boundaries and radii by introducing a small number of samples, and obtains interval parameters under a given confidence level. On the basis of uncertainty quantification, the dynamic reliability of the vibration system is derived, and nonprobabilistic reliability is introduced into the design of active controllers through optimization strategies. The reliable control strategy aims to select the optimal controller parameters on a reliable path with a given credibility, in order to solve the problem of overly conservative robust control theory. Two numerical examples were used to verify the feasibility and applicability of this method.

Implementing Reliability Analysis of a Soil Slope using First-Order Second Moment Method, Monte-Carlo Simulation Method and Subset Simulation Method in a MS-Excel Spreadsheet

The spatial variation in the soil properties affects the analysis and design of geotechnical structures such as slope stability analysis. This paper presents the reliability analysis of a soil slope using three different methods namely, first-order second moment (FOSM) method, Monte-Carlo simulation (MCS) method and Subset simulation (SS) method. The analysis has been carried out by considering the spatial variation of the soil properties of the slope. The correlation function and the correlation distance (λ) for the uncertain parameters are modelled using random field theory developed by Vanmarcke. The factor of safety (FOS) of the slope is determined using Ordinary method of slices. The reliability based probabilistic slope stability analysis has been carried out in a MS-Excel spreadsheet to determine the probability of failure (P_f ) and reliability index (β) of the slope. The spreadsheet mainly consists of three parts i.e., deterministic analysis, uncertainty analysis and uncertainty propagation. The study shows that the SS method has shown better performance as compared to FOSM method and MCS method especially at low failure probability levels. Also, SS method ensures the generation of samples in the failure region which is not always possible in MCS method.

Asymptotically matched extrapolation of fishnet failure probability to continuum scale

Motivated by the extraordinary strength of nacre, which exceeds the strength of its fragile constituents by an order of magnitude, the fishnet statistics became in 2017 the only analytically solvable probabilistic model of structural strength other than the weakest-link and fiber-bundle models. These two models lead, respectively, to the Weibull and Gaussian (or normal) distributions at the large-size limit, which are hardly distinguishable in the central range of failure probability. But they differ enormously at the failure probability level of 10−6, considered as the maximum tolerable for engineering structures. Under the assumption that no more than three fishnet links fail prior to the peak load, the preceding studies led to exact solutions intermediate between Weibull and Gaussian distributions. Here massive Monte Carlo simulations are used to show that these exact solutions do not apply for fishnets with more than about 500 links.The simulations show that, as the number of links becomes larger, the likelihood of having more than three failed links up to the peak load is no longer negligible and becomes large for fishnets with many thousands of links. A differential equation is derived for the probability distribution of not-too-large fishnets, characterized by the size effect, the mean and the coefficient of variation.Although the large-size asymptotic distribution is beyond the reach of the Monte Carlo simulations, it can by illuminated by approximating the large-scale fishnet as a continuum with a crack or a circular hole. For the former, instability is proven via complex variables, and for the latter via a known elasticity solution for a hole in a continuum under antiplane shear. The fact that rows or enclaves of link failures acting as cracks or holes can form in the large-scale continuum at many random locations necessarily leads to the Weibull distribution of the large fishnet, given that these cracks or holes become unstable as soon they reach a certain critical size. The Weibull modulus of this continuum is estimated to be more than triple that of the central range of small fishnets. The new model is expected to allow spin-offs for printed materials with octet architecture maximizing the strength–weight ratio.

AK-SEUR: An adaptive Kriging-based learning function for structural reliability analysis through sample-based expected uncertainty reduction

Reliability Analysis (RA) is a critical aspect of structural design and performance evaluation aiming to determine the probability of structural failure under given random input parameters. With modern development of modeling techniques, computational models have achieved higher fidelity but at the increased cost of computational time, which poses a significant challenge for RA. Consequently, surrogate model-assisted RA has been explored as a means of improved efficiency and accuracy. This study proposes a novel learning function, Sample-based Expected Uncertainty Reduction (SEUR), for surrogate model-assisted RA. The SEUR function uses statistical information from the metamodeling with fixed hyper-parameters to construct expected failure probability bounds to sequentially update the design of experiment (DoE). The joint probability densities of input variables are accounted for through simulation methods, including Monte Carlo (MC) and subset simulation (SS). Furthermore, the discrete simulated annealing algorithm is used to search for the optimal design point. The performance of proposed AK-SEUR function is systematically evaluated using six examples of different dimensions, failure probability levels and nonlinearities. The AK-SEUR function is demonstrated to be more effective and efficient than other popular active learning methods in dealing with nonlinear performance functions, small probabilities, and complex limit states. The proposed SEUR function has the potential to improve the efficiency and accuracy of RA, particularly in situations where computational models are time-consuming and the search for the optimal solution is challenging.

Improvement of integrity management for pressure vessels based on risk assessment - A natural gas separator case study

The process of oil and gas processing plant is complex, the types of pressure vessels are rich, and the functions are critical. However, the working medium is mostly untreated medium, and the hazard factors are complex, which poses a threat to the safe production of oil and gas processing plant. Based on PDCA cycle, this paper establishes a six-step links of integrity management for sustainable improvement of pressure vessels. The typical failure modes of pressure vessels are determined, and the fishbone diagram of risk factors under each failure mode is compiled. Risk quantification and classification of pressure vessels based on failure modes (RBFM) is innovatively proposed. Avoiding incalculable failure frequency index, the process quantification of failure possibility is formed according to the development of hazard factors. A failure consequence calculation model based on the leakage affected area was established. Combined with the failure probability level and risk level, the hierarchical inspection strategy for pressure vessels under different failure modes is established. Finally, the method is applied to the natural gas separator of H processing plant. The research results show that RBFM proposed in this paper can meet the requirements for rapid and accurate risk assessment of pressure vessels in oil and gas processing plant. This paper establishes a safe production barrier for the pressure vessel and improves the intrinsic safety of the equipment.

Copula‐based importance sampling method for efficient slope reliability analysis

AbstractEfficient slope reliability analysis that accounts for the interdependencies of random variables is challenging. In this study, an efficient reliability method is proposed based on importance sampling (IS) and copulas. First, the IS method based on the design point is introduced to construct the marginal IS density. Copulas are subsequently adopted to capture the interdependencies and establish both the initial joint distributions and the joint IS densities of random variable pairs. The IS algorithm is modified to consider the dependencies through not only the estimation of the probability distribution but also random sampling. Finally, a framework of the copula‐based IS method is constructed. The applicability, accuracy, efficiency, and robustness of the proposed method were validated by a classical slope case. The results showed that with the proposed method, slope reliability could be efficiently assessed with a relatively high accuracy. The dependencies of the variables significantly affected the IS and final slope reliability, whereas the IS density had a significantly lower effect. The proposed method combines the advantages of IS and copulas well. Interdependencies of the variables can be properly characterized during sampling. Moreover, the proposed method was shown to be robust regardless of the system failure probability level. It can be further applied to improve the random finite element method in the future.

Dynamic risk assessment for underground gas storage facilities based on Bayesian network

Loss of the underground gas storage process can have significant effects, and risk analysis is critical for maintaining the integrity of the underground gas storage process and reducing potential accidents. This paper focuses on the dynamic risk assessment method for the underground gas storage process. First, the underground gas storage process data is combined to create a database, and the fault tree of the underground gas storage facility is built by identifying the risk factors of the underground gas storage facility and mapping them into a Bayesian network. To eliminate the subjectivity in the process of determining the failure probability level of basic events, fuzzy numbers are introduced to determine the prior probability of the Bayesian network. Then, causal and diagnostic reasoning is performed on the Bayesian network to determine the failure level of the underground gas storage facilities. Based on the rate of change of prior and posterior probabilities, sensitivity and impact analysis are combined to determine the significant risk factors and possible failure paths. In addition, the time factor is introduced to build a dynamic Bayesian network to perform dynamic assessment and analysis of underground gas storage facilities. Finally, the dynamic risk assessment method is applied to underground gas storage facilities in depleted oil and gas reservoirs. A dynamic risk evaluation model for underground gas storage facilities is built to simulate and validate the dynamic risk evaluation method based on the Bayesian network. The results show that the proposed method has practical value for improving underground gas storage process safety.

Flexure fatigue strength and failure probability of self compacting concrete made with RCA and blended cements

The examination of the fatigue strength of self-compacting concrete (SCC) under flexural loads is the prime objective of the present investigation. SCC comprises recycled concrete aggregates (RCA) and blended cements, i.e., either silica fume (SF) or metakaolin (MK), which differentiates it from the reference mix. Fatigue life data has been accumulated by conducting flexural fatigue tests on beam specimens of size 100 mm × 100 mm × 500 mm. The existing fatigue equations based on SN relationships have been tested with the experimental data and proposed to predict the flexural fatigue strength for SCC constituting RCA and blended cements. An approach based on failure probability (Pf) has also been adopted for the prediction of flexural fatigue strength of SCC mixes. Therefore, SNPf curves have been generated for various SCC mixes under consideration. Several mathematical equations have also been proposed for various SCC mixes to predict the flexure fatigue strength at the desired level of failure probability (Pf). Endurance limit has also been evaluated for various SCC mixes. The introduction of blended cements in SCC mixes constituting RCA enhances the endurance limit, but the enhancement of endurance limit is most prominent in SCC mix containing RCA and MK.

Statistical analysis of S[sbnd]N type environmental fatigue data of Ni-base alloy welds using weibull distribution

In this study, the probabilistic fatigue life model for Ni-base alloys was developed based on the Weibull distribution using statistical analysis of fatigue data reported in NUREG/CR-6909 and the new fatigue data of Alloy 52M/152 and 82/182. The developed Weibull model can consider right-censored data (i.e., non-failed data) and quantify the improved safety (or reliability) based on the level of failure probability. The overall margin in the current fatigue design limit model (ASME design curve + NUREG/CR-6909 Fen model) is similar to that of the Weibull model with a cumulative failure probability of approximately 2.5%. The margin in the current fatigue design limit model demonstrated inconsistencies for the Ni-base alloy weld data, whereas the Weibull model showed a consistent margin. Therefore, the Weibull model can systematically mitigate the excessive safety margin.

Sufficient conditions for equivalence between safety factor-based and reliability-based design requirements

Although there is a perceptible trend of developing and adopting reliability-based design (RBD) in geotechnical engineering, geotechnical practitioners are used to safety factor-based design (SFBD) due to its simplicity. At the stage of transition from SFBD to RBD, geotechnical practitioners frequently compare RBD results with SFBD results that they get used to for cross-checking. Nevertheless, SFBD and RBD, in theory, may provide different design outcomes because they use different design methodologies and requirements. The equivalence between SFBD and RBD (i.e., with the same feasible designs) can only be achieved under some conditions. This study proposes sufficient conditions for achieving the equivalence between SFBD and RBD based on the probabilistic sufficiency factor (i.e., a fractile of the performance function at the target failure probability level PTE). Simplified verification methods of the proposed sufficient conditions in three simple cases are provided. The proposed sufficient conditions and their verification are illustrated using a gravity retaining wall example with a separable performance function and a deep excavation example with an inseparable performance function. Results demonstrate the validity of the proposed sufficient conditions, under which a unique mapping from the allowable factor of safety to PTE for a given design situation can be established.

Estimation of Reservoir Volumes at Drafts of 40% - 90%: Drought Magnitude Method Applied on Monthly River Flows from Canadian Prairies

The draft ratios for sizing the reservoirs can vary within a wide range (40% - 90% of the mean annual flow, MAF), depending upon the demands for water by various users, and environmental and ecological considerations. The reservoir volumes based on the drought magnitude (DM) method were assessed at aforesaid draft ratios using monthly-standardized hydrological index (SHI) sequences of 10 Canadian rivers located in the Canadian prairies and northwestern Ontario. These rivers are typified by a high level of persistence lag-1 autocorrelation, ρ1m ≥ 0.50 and up to 0.94) and coefficient of variation (cvo) in the range of 0.42 to 1.48. The moving average (MA) smoothing of monthly SHI sequences formed the basis of the DM method for estimating reservoir volumes. The truncation or cutoff level in the SHI sequences was found as SHIx [=(α - 1)μo/σo], [(α - 1)μo/σmax], or [(α - 1)μo/σav], where α (=0.40 to 0.90) is the draft ratio i.e. proportion of the MAF, μo and σo are the overall mean and standard deviation of the monthly flows, σmax is the maximum value of standard deviations and σav the average of 12 monthly values. The failure probability levels (PF) were fixed at 5%, 2.5% and 0% (corresponding reliability of 95%, 97.5% and 100%). The study revealed that the coefficient of variation is the most important parameter that influences the reservoir size while the role of lag-1 autocorrelation (ρ1m) appears more pronounced at high draft ratios, α such as 0.90, 0.80 and 0.70 in increasing the reservoir size. The DM based method can be regarded as an alternative to Behavior analysis for sizing reservoirs at the desired probability of failure or reliability level.

Seismic reliability evaluation of bridges under spatially varying ground motions using a four-parameter distribution

Large-span bridges are often subjected to spatially varying ground motion (SVGM) excitations. Meanwhile, bridge structures and ground motion excitations generally present a large degree of inherent uncertainties. Carrying out the seismic reliability analysis of bridges under SVGMs can provide a direct assessment for the safety probability of bridges during an earthquake event and is of paramount importance for the probabilistic performance analysis of bridges. To this end, this study proposes a new method for the seismic reliability evaluation of random nonlinear bridges subjected to SVGMs, which tackles the drawback of the current seismic reliability analysis method that can only be applied for uniform seismic excitations. To implement, the space correlated ground motions are first generated via a random function-based spectral representation method, based on which the uncertainty of ground motions is characterized by three elementary random variables. The first-passage probability of bridges under SVSGMs is then derived by the equivalent extreme value distribution (EVD), which is approximated by a flexible four-parameter distribution, i.e., shifted generalized lognormal distribution (SGLD). To estimate the parameters in the SGLD model with high efficiency and accuracy, a novel fractional moment-based method is proposed. A numerical example of a two-span continuous box-girder bridge subjected to space varying site conditions is investigated to verify and exemplify the proposed method. The results demonstrate that: 1) The proposed method is efficient and accurate for the seismic reliability of bridge under SVGM, even at low failure probability levels, and the computational burden of the proposed method is less than 2% of that of MCS; 2) The spatial variability is significantly influential on the seismic reliability of the bridge, ignoring the spatial variability of seismic excitation will significantly underestimate the failure probability of the bridge.

A failure boundary exploration and exploitation framework combining adaptive Kriging model and sample space partitioning strategy for efficient reliability analysis

Surrogate model-based methods have gradually become a vital method to assess reliability. However, the existing methods usually ignore the memory problems of matching candidate samples with the level of failure probability, which leads to inefficiency and even restricts their applicability. Therefore, this work combining the adaptive Kriging model and sample space partitioning strategy proposes a failure boundary exploration and exploitation framework (FBEEF), which divides the construction process of the adaptive Kriging model into two phases using different candidate samples to enrich training samples. In the exploration phase, a sample space partitioning strategy combining K-means clustering and slice sampling is employed to obtain several subsets and static candidate samples. In the exploitation phase, the approximate distances between the static candidate samples and the failure boundary are calculated to identify important subsets, whose samples are named dynamic candidate samples. Furthermore, a new stopping criterion is developed by combining leave-one-out method and weighted simulation method. To improve the efficiency of FBEEF Monte Carlo simulation or Importance Sampling is selected to estimate the final failure probability. Five examples were analyzed to test the effectiveness of FBEEF, and the results show that FBEEF can obtain good results with fewer training samples and lower analysis time.

A new reliability method for small failure probability problems by combining the adaptive importance sampling and surrogate models

Reliability analysis for structural systems with multiple failure modes and expensive-to-evaluate simulations is challenging. In this paper, a new and efficient system reliability method is proposed based on the adaptive importance sampling and kriging models. The Metropolis–Hastings (M–H) algorithm is used to construct several Markov chains to fully explore complex failure regions. A number of Markov chain states are selected as the center of the component importance sampling functions to generate samples for reliability analysis. Based on the component importance sampling function of each selected chain state, the system importance sampling function is constructed with the weighting index. The system importance sampling function can be constructed effectively because it does not involve time-consuming simulations and the most probable point (MPP) search. The new learning function, which is directly linked to the system failure probability, is developed to adaptively select the best added samples for refining the kriging models at each iteration. The adaptive importance sampling method and kriging models are well-combined for system reliability analysis in the proposed method. Compared with existing methods, the proposed method, generally, offers the following advantages: (1) The learning function and stopping criterion are directly linked to system failure probability; (2)the adaptive importance sampling and kriging models are well-combined to yield accurate results based on a small sample size for small failure probability problems; (3) the weights of sampling centers are considered, and the MPP search is not required at each iteration; (4) it is applicable for complex systems regardless of the structure and system failure probability level. Three numerical examples are analyzed, which demonstrate that the proposed method is effective for complex system reliability analysis.

An Analytical Random Field Solution for the Reliability of Axially Loaded Piles in the Ultimate Limit State Considering the Effect of Soil Sampling

The present work is concerned with the effect of soil spatial variability on estimating the ultimate soil resistance of floating axially loaded piles from point measurements of soil properties along the pile. The ultimate limit state is considered. In particular, closed form formulae for (i) determining the optimal sampling depth for minimizing statistical uncertainty and (ii) the optimal—minimum required—safety factor for a desired failure probability level are derived. A dimensionless parameter, the cohesion-to-friction parameter Λ, is introduced which quantifies the weight of soil’s cohesion contribution relative to that of soil’s friction in the linear trend of the ultimate soil strength. The analysis shows that the probability of failure profile with the sampling depth attains a minimum, designating the optimal sampling point. This depends on the scaled spatial correlation length of the soil Θ (i.e., the spatial correlation length of soil over the length of the pile) and the parameter Λ, but not on the coefficient of variance of the ultimate soil strength (covu) or the safety factor. Furthermore, it was found that the optimal depth is always at the lower half of the pile, approaching the mid-point or the bottom end of the pile for Λ>>1 or Λ<<1, respectively. In addition, it was found that for large Θ, the optimal depth tends to be closer to the mid-point of the pile, while for small Θ, the optimal sampling depth arises closer to the bottom end. The practical usefulness of the results is related to a suitable choice of the safety factor. Inverting the probability of failure formula at its minimum value, an optimal safety factor is obtained as a function of the desired (acceptable) probability of failure, and the parameters Θ, Λ and covu. The optimal safety factor is the minimum value required for a desired level of the probability of failure.

General Fishnet Statistics of Strength: Nacreous, Biomimetic, Concrete, Octet-Truss, and Other Architected or Quasibrittle Materials

Abstract The fishnet probabilistic model was recently developed to characterize the strength distribution of nacre-like biomimetic materials. It reveals that the unique fishnet-like connectivity of the material microstructure brings about enormous safety gain at the extremely low failure probability level of one out of a million, desired for engineering structures. The gist of the theory is that the material microstructure plays a determining role in its failure probability tail. Therefore, a carefully designed connectivity for a material microstructure not only enhances its mean strength but also significantly reduces its marginal failure risk. Here, we first show that the initially introduced series expansion and the newer formulation based on order statistics are, in the fishnet model, essentially equivalent. From that we develop a neat general form of the fishnet statistics. Then, we extend our theoretical approach to the strength distributions of architected nanomaterials such as the printed octet-truss carbon nanolattices, as well as to quasibrittle particulate composites such as concrete, and formulate a unified general fishnet statistics. We demonstrate that the octet-truss system can be physically seen and statistically treated as a union of three fishnets with three mutually orthogonal orientations. We show that the three-dimensional assembly of fishnets further enhances the tail strength at the 10−6 probability quantile, compared to two-dimensional (2D) fishnet statistics. We compare the performance of different statistical strength models by fitting of the simulated and experimental histograms data for the octet-truss nanolattice. Finally, we argue that, at the extreme lower tail of failure probability, quasibrittle materials such as concrete or fiber composites should partially exhibit the fishnet-type statistical behavior.

Subset simulation for efficient slope reliability analysis involving copula-based cross-correlated random fields

This study proposes a subset simulation (SS)-based approach for efficient slope reliability analysis involving copula-based cross-correlated random fields of cohesion (c) and friction angle (ϕ) of soils. First, the copula theory for modeling the cross-correlation between c and ϕ is briefly introduced. The algorithms for generating the copula-based cross-correlated random fields of c and ϕ are detailed. Then, the SS for efficient slope reliability analysis involving copula-based cross-correlated random fields of c and ϕ is explained. Finally, two slope examples with the same geometry but different sources of probability information are presented to illustrate and demonstrate the proposed approach. The results indicate that the proposed approach has both good accuracy and efficiency in slope reliability analysis involving the copula-based cross-correlated random fields of c and ϕ at low failure probability levels. The copula theory for characterizing the cross-correlated random fields can consider both the Gaussian and non-Gaussian dependence structures between c and ϕ. The copula selection has a significant impact on slope reliability with spatially variable c and ϕ. The probabilities of slope failure produced by different copulas differ considerably. This difference increases with decreasing probability of slope failure. The commonly-used Gaussian copula may lead to a significant underestimate of the probability of slope failure. The reasonable identification of the best-fit copula for characterizing the cross-correlated random fields of c and ϕ based on the test data is highlighted in practical slope reliability analysis.

Chaotic enhanced colliding bodies optimization algorithm for structural reliability analysis

It is of extreme importance to assess the failure probability and safety level of structural system in structural design. Nowadays, many researchers presented several approaches for structural reliability analysis, such as the first-order reliability method, Monte Carlo simulation, and the meta-heuristic algorithm. The meta-heuristic algorithm is not only efficient to solve global optimization problems but also shown to be an effective tool for structural reliability analysis. A recent meta-heuristic optimization approach, enhanced colliding bodies optimization, has emerged as a relatively simple implementation with a fast convergence speed. Chaos theory is characterized by its ergodicity, pseudo-randomness, and irregularity. This article thus presents a novel approach introducing chaotic maps into the enhanced colliding bodies optimization algorithm to promote the performance of convergence, named as chaotic enhanced colliding bodies optimization algorithm. The proposed algorithm uses chaotic maps to change the generation pattern of particles and improve convergence characteristics. A procedure based on the effective use of the represented chaotic enhanced colliding bodies optimization is then applied in structural reliability analysis. A variety of numerical and structural problems are tested in this article to demonstrate that the given method actually improves the performance of enhanced colliding bodies optimization in convergence as well as the accuracy for reliability analysis compared with the other methods existing in the literature.

An Effective Approach for Reliability-Based Sensitivity Analysis with the Principle of Maximum Entropy and Fractional Moments.

The reliability-based sensitivity analysis requires to recursively evaluate a multivariate structural model for many failure probability levels. This is in general a computationally intensive task due to irregular integrations used to define the structural failure probability. In this regard, the performance function is first approximated by using the multiplicative dimensional reduction method in this paper, and an approximation for the reliability-based sensitivity index is derived based on the principle of maximum entropy and the fractional moment. Three examples in the literature are presented to examine the performance of this entropy-based approach against the brute-force Monte-Carlo simulation method. Results have shown that the multiplicative dimensional reduction based entropy approach is rather efficient and able to provide reliability estimation results for the reliability-based sensitivity analysis of a multivariate structural model.

Time-variant reliability of timber beams according to Eurocodes considering long-term deflections

ABSTRACT In order to achieve a consistent level of failure probability, structural design codes are optimized using probabilistic methods. This optimization process traditionally focuses on the ultimate limit states (ULS). However, in the design of timber structures the performance of the structural members is often governed by the serviceability limit state (SLS) associated with different load levels than applied in the ULS. The probability of serviceability failure is strongly dependent on the loading sequence and the time-dependent response of timber; therefore, a time-variant probabilistic model is recommended to estimate them properly. This study aims to investigate the time-dependent reliability for long-term deflections of timber office and residential floor beams according to the specifications of the Eurocodes. A simple creep model is used to calculate the deflections and Monte Carlo simulation is carried out to determine the reliability index. It was found that the creep factor and the suggested deflection limits given in Eurocode 5 might not be appropriate to achieve the expected target reliabilities. To obtain a more consistent reliability, more suitable values for the mentioned parameters were suggested. However, the primary aim was to present a framework to determine appropriate deflection limits for structural codes.