Second-order Second-moment Research Articles

Chatter reliability of high speed cylindrical grinding with uncertain parameters

Cylindrical grinding is commonly used as the final process of machining, but its quality and efficiency are often limited by chatter phenomenon. Many research show that the result from conventional chatter analysis by using the deterministic parameters is not consistent with the practical working condition and the chatter reliability prediction in high speed cylindrical grinding with uncertain parameters is studied in this paper. The dynamic model of the cylindrical grinding system is established, and uncertain parameters such as mass, stiffness and damping coefficient are introduced, and the stability lobe diagrams with uncertain parameters were obtained. The reliability model of the grinding system was established, and the reliability was solved by using the Monte Carlo simulation method, the improved first-order second-moment method and the second-order second-moment method, respectively. Then, the chatter-free reliability of the system was predicted by using the reliability lobe diagram. The accuracy of the grinding reliability prediction results is verified by the experiment results. Furthermore, the Monte Carlo sensitivity analysis method is used to study the influence of random parameters on the grinding system, and the results show that the variation of modal mass has the greatest influence on the reliability of the grinding system.

Reliability analysis of an inter-story isolated structure under a main-aftershock sequence based on the Laplace asymptotic method

After a strong mainshock, subsequent ground motion is the result of a sequence of multiple aftershocks, and the damage to a structure under these conditions is more severe than from a single earthquake. Most seismic studies are based on a single earthquake event. To explore the influence of a main-aftershock sequence on an isolated inter-story structure, we constructed a three-dimensional finite-element model of such a structure and subjected it to repeated main-aftershock sequences. The Laplace asymptotic method of second-order second-moment was used to calculate the reliability of the structure under the action of a single mainshock and after a main-aftershock sequence at different seismic levels. The effects of the number of aftershocks, the location of the isolation layer, and the stiffness of the isolation bearing in the structure were analyzed. The results showed that aftershocks increased the failure probability of each sub-structural part of the inter-story isolated structure. The failure probability of the lower structure had the greatest influence, which was about 3.89 times that for the mainshock alone. The probability of failure from multiple vs single aftershocks was similar, but the magnitude of the aftershock plays a major role in failure. The number of aftershocks reduced the overall reliability of an inter-story isolation structure. In the case of different isolation layer positions, the placement of the isolation layer at the top of the seventh story under an extremely rare earthquake level resulted in a reduction of 6.01%. With isolation bearings of different stiffness, the largest decrease was 7.88% when the stiffness was 50%.

Epistemic Uncertainty Treatment Using Group Method of Data Handling Algorithm in Seismic Collapse Fragility

Abstract Developing fragility functions is the essential step in incorporating important uncertainties in next-generation performance-based earthquake engineering (PBEE) methodology. The present paper is aimed to involve record-to-record variability as well as modelling uncertainty sources in developing the fragility curves at the collapse limit state. In this article, in order to reduce the dispersion of uncertainties, Group Method of Data Handling (GMDH) in combination with Monte Carlo Simulation (MCS) is used to develop structural collapse fragility curve, considering effects of epistemic and aleatory uncertainties. A Steel Moment Resisting Frame (SMRF) is chosen as the tested structure. The fragility curves obtained by the proposed method which belongs to GMDH approaches are compared with those resulted from simple and well-known available methods such as First-Order Second-Moment (FOSM), Approximate Second-Order Second-Moment (ASOSM) and Monte Carlo (MC)/Response Surface Method (RSM), somehow, as an accurate method. The results of the application of the proposed approach indicate increasing accuracy and precision of the outputs as well as power with the same computational time compared to aforementioned methods. The GMDH method introduced here can be applied to the other performance levels.

Reliability-Based Topology Optimization Using Mean-Value Second-Order Saddlepoint Approximation

A reliability-based topology optimization (RBTO) approach is presented using a new mean-value second-order saddlepoint approximation (MVSOSA) method to calculate the probability of failure. The topology optimizer uses a discrete adjoint formulation. MVSOSA is based on a second-order Taylor expansion of the limit state function at the mean values of the random variables. The first- and second-order sensitivity derivatives of the limit state cumulant generating function (CGF), with respect to the random variables in MVSOSA, are computed using direct-differentiation of the structural equations. Third-order sensitivity derivatives, including the sensitivities of the saddlepoint, are calculated using the adjoint approach. The accuracy of the proposed MVSOSA reliability method is demonstrated using a nonlinear mathematical example. Comparison with Monte Carlo simulation (MCS) shows that MVSOSA is more accurate than mean-value first-order saddlepoint approximation (MVFOSA) and more accurate than mean-value second-order second-moment (MVSOSM) method. Finally, the proposed RBTO-MVSOSA method for minimizing a compliance-based probability of failure is demonstrated using two two-dimensional beam structures under random loading. The density-based topology optimization based on the solid isotropic material with penalization (SIMP) method is utilized.

A reliability analysis method for rock slope controlled by weak structural surface

Catastrophic landslides maybe occur in rock slope due to the effect of strong earthquakes or heavy rainfall. The stability of rock slope is usually controlled by different scales of weak structural surfaces, which are uncertain and randomly exist in the rock slope. According to the geological characteristics of rock slope, two typical failure modes – plane and wedge are possible. A second-order second-moment (SOSM) method is presented to calculate the reliability index and the failure probability of rock slope, which is an improvement over the first-order second-moment (FOSM) method, and performance functions are built up with the classic limit equilibrium method. The presented method is applied to analyze the failure probability of two rock slopes at the Jinping I Hydropower Station and is compared with the Monte Carlo method and the FOSM method. The computed results show that for plane failure, the reliability index and the failure probability determined by the presented method are 0.563 and 28.7%, respectively, and the reliability index and the failure probability determined by Monte Carlo method are 0.677 and 24.9%, respectively. However, for the FOSM method, the reliability index and failure probability are –0.025 and 51.0%, respectively. For both plane failure and wedge failure, the difference between the presented method and the Monte Carlo method is very small, but the failure probability of plane failure determined by FOSM method is larger than that of the other two methods. The presented method can provide a useful tool to evaluate the failure probability of rock slope.

Settlement predictions of footings on sands using probabilistic analysis

The design of footings on sands is often controlled by settlement rather than bearing capacity. Therefore, settlement predictions are essential in the design of shallow foundations. However, predicted settlements of footings are highly dependent on the chosen elastic modulus and the used method. This paper presents the use of probabilistic analysis to evaluate the variability of predicted settlements of footings on sands, focusing on the load curve (predicted settlements) characterization. Three methodologies, the first- and second-order second-moment (FOSM and SOSM), and Monte Carlo simulation (MCS), for calculating the mean and variance of the estimated settlements through Schmertmann (1970)'s equation, are presented and discussed. The soil beneath the footing is treated as an uncorrelated layered material, so the total settlement and variance are found by adding up the increments of the layers. The deformability modulus (ESi) is considered as the only independent random variable. As an example of application, a hypothetical case of a typical subsoil in the state of Espirito Santo, southeast of Brazil, is evaluated. The results indicate that there is a significant similarity between the SOSM and MCS methods, while the FOSM method underestimates the results due to the non-consideration of the high-order terms in Taylor's series. The contribution of the knowledge of the uncertainties in settlement prediction can provide a safer design.

Reliability-based analysis and design of steel frames with different supports

Purpose – The behaviours of four indeterminate frame-support combinations namely Type I (with fixed supports), Type II (with pinned supports), Type III (with fixed-pinned supports) and Type IV (with fixed-roller supports) frames under the exposure conditions and loads as existing on site were simulated. Two categories of these combinations (I and II) were studied namely single storey-single bay and multiple storey-single bay frames, as illustrated in the case studies treated. A procedure for determining the probability of failure at different sections along the frame types, the range between the probability of failure bounds and the reliability ratings of the frame types were developed based on the kinetic method of plastic moment analysis, minimum weight design method, piecewise method of moment analysis and first order-second moment (FOSM) methods. The analysis results of the Category I frames showed that the Type I frame was most reliable (with the lowest probability of failure range of 0.3269), while the Type II frame was least reliable (with the highest probability of failure range of 0.4918). These results were consistent with those of the Category II frames. The paper aims to discuss these issues. Design/methodology/approach – Collapse mechanisms were generated for four frame-support types and the corresponding plastic moments were determined using both the kinematic plastic analyses and minimum weight design methods. The members were designed and the plastic moments were distributed at sections of constant interval along the frame length to generate corresponding envelopes. A similar process was carried out to determine the elastic moment variables due to the loads. The reliability index and the corresponding probability of failure at each frame section were determined. Then, the probabilities of failure bounds for the frames were then compared to determine the most reliable. Findings – It was observed that there existed a wide margin between the elastic and plastic moments indicating that design of steel structures at the elastic limit does not take full advantage of its strength. Hence, the design can be carried out beyond the elastic limit and within the safety margin given in equation (3). However, the safety of the entire frame is assessed on the basis of range of values between the highest and the lowest probability of failure bounds. The lower this range is (not exceeding 0.5 or 50 per cent), the more reliable the frame is. Research limitations/implications – The equations developed in this study can only be directly applied to multi storey-single bay frames. However, the reliability-based analysis and design procedure developed can be extended to other types of frames. Practical implications – A practical approach for analysing steel frames with different supports with the overall goal of producing safe and economical designs has been developed and presented in this paper. Originality/value – The procedure adopted is very original and can be backed up by existing literature. The piecewise method for analysing moments at various sections along a frame is also innovative. The whole concept can be adopted to determine the reliability of other types of frames such as multiple bay-multistorey frames with different support types.

Reliabilty-based Torsional Design of Reinforced Concrete Beams Strengthened with CFRP Laminate

The structural reliability of torsional concrete beams strengthened with full and strip wrapping of carbon fiber reinforced polymer (CFRP) laminate is investigated. The first order-second moment reliability method has been applied to make a reliability assessment on the torsional capacity designed by technical guideline in Iran. The code No.345 published by the Management and Planning Organization of Iran (MPO 345-2006) is the first technical code in Iran for design of concrete members strengthened with CFRP laminate. In this work, the average reliability indexes for unstrengthened, strengthened with full wrap and strengthened with strip wrap beams have been found. The results indicate that the MPO design guideline are some unconservative. A reliability strengthening ratio is introduced for assessment of the variation in average reliability index before and after strengthening with different resistance factors. A parametric study on this factor reveals that if the reliability level of the strengthened reinforced concrete beams is kept to be consistent with their similar unstrengthened beams, a value of 0.89 and 0.81 for strengthened with full wrap and strengthened with strip wrap, should be applied, respectively.

Probabilistic assessment of the structural failure of single-lap bonded joints

Single-lap bonded joints are a common typology, frequently used when designing structural systems. Several authors have studied their behaviour in different situations, involving both failure of the adhesive and also of the adherends. So far, the design of such joints has been based on deterministic postulates. In this work, a non-deterministic approach is followed. The reliability of the joint is assessed with several limit state approximation methods. Furthermore, sampling methods and a global reliability approach are considered. The algorithms are applied to the reliability analysis of a generic single-lap bonded joint, where the specific uncertainties related with this typology are taken into account, including mechanical properties and geometrical parameters of adherends and adhesive. All of them are defined as random variables and different cases are studied. Results show that first- and second-order second-moment and advanced mean value methods give an estimation to calculate the probability of failure at early stages of design, when precise values are not required. However, methods based on approximations of limit state surface around the most probable point of failure present a better response in terms of accuracy. Global reliability methods offer a good alternative to limit state approximations, as they present similar accuracy and usually require fewer iterations. Sampling methods often need extensive use of computational resources and are suitable for those cases with a large number of random variables, where computational cost is comparable with that required by other methods.

신뢰성이론에 기반한 해양환경 콘크리트의 내구수명 평가

Monte-Carlo simulation technique is often used in order to predict service life of concrete structure subjected to chloride penetration in marine environment based on probability theory. Monte-Carlo simulation method, however, the method gives different results every time that the simulation is run. On the other hand, moment method, which is frequently used in reli- ability analysis, needs negligible computational cost compared with simulation technique and gives a constant result for the same problem. Thus, in this study, moment method was applied to the calculation of corrosion-initiation probability. For this purpose, computer programs to calculate failure probabilities are developed using first-order second moment (FOSM) and second-order sec- ond moment (SOSM) methods, respectively. From the analysis examples with the developed programs, SOSM was found to give a more accurate result than FOSM does. The sensitivity analysis has shown that the factor affecting the corrosion-initiation prob- ability the most was the cover depth, and the corrosion-initiation probability was influenced more by its coefficient of variation than its mean value.

Tractable Conflict Risk Accumulation in Quadratic Space for Autonomous Vehicles

Recent developments have shown that conflict avoidance may be cast as a probabilistic state-space problem. The probabilistic accumulation of risk because of exposure to a hazard over a period of time is an essential and often neglected and misunderstood aspect of the process of conflict risk calculation. How such risk accumulation may be accurately approximated in real time is demonstrated. The research reverts to theoretical basics and illustrates how risk accumulation for a one- and two-vehicle scenario is a first passage time problem. It is illustrated how exact quadratic state-order reduction may be combined with simple and accurate first passage time approximations. In this way, a tractable and accurate receding horizon solution is provided with the recognition of the time dependence of risk accumulation. The solution algorithm is implemented as a relatively straight-forward four-step discrete-time propagation and accumulation of risk. Small calculation errors are characterized to first order-second moment and always lead to a slight upper-bound solution, thereby improving safety. Simulation results indicate a high degree of risk calculation accuracy, calculable in real time.

Second-Order Third-Moment Reliability Method

In the second-order reliability method (SORM), the failure probability is generally estimated using a parabolic approximation of a performance function. In the present paper, the moment properties of a second-order approximation of performance functions are investigated, and a moment approximation for a second-order reliability method and a simple second-order third-moment reliability index are proposed for the estimation of failure probability corresponding to both the simple and general parabolic approximations. Based on the property that the parabolic approximation approaches a unit normal random variable, the ranges of three parameters are investigated: the number of variables, the principal curvature, and the first-order reliability index. A simple analytical judgment formula is derived, which can help us judge when the first-order reliability method is sufficiently accurate and when the SORM is required. A simple second-order second-moment reliability index is also proposed for problems with relatively small principal curvatures. Through some numerical examples, the simplicity and accuracy of the second-order second- and third-moment reliability indices are demonstrated.

Stochastic finite element method applied to non-linear analysis of embankments

The deterministic Finite Element Method (FEM) is a valuable tool for understanding and predicting the mechanical behaviour of earth structures. The main difficulty in the application of this technique generally arises from the large uncertainties affecting the mechanical properties of materials to be introduced in the analysis. In many instances, these parameters should actually be considered as random variables or random fields. The Stochastic Finite Element Method (SFEM) should then be used to assess the results of the analyses in probabilistic terms. In this paper, the usefulness of the SFEM approach for engineering purposes is discussed and illustrated by analyses of embankments constructed by placing successive lifts of compacted soil. Construction materials are assumed to follow a simple non-linear constitutive law (Duncan JM, Chang CY. Non-linear analysis of stress and strain in soils, Journal of the Soils Mechanics and Foundation Division, ASCE 1970;96(5):1629–1653). Stochastic finite element analyses are performed using both the First Order-Second Moment method (FOSM) and Monte Carlo simulations (MC). A simple example shows that SFEM analyses can be useful to evaluate the relative influence of each of the parameters of the constitutive model on the results. Uncertainties affecting displacements, strains and stresses predictions for a large earth dam are also presented.

Evaluation of web buckling test results on welded beams and plate girders subjected to shear

The limit state design requirements for welded beams and plate girders in shear, and combined bending and shear, are analysed and criteria for two codes (AISC and Eurocode 3) are compared. As the differences in the design requirements are substantial, a comparison of the achieved reliability indices is made in order to find which model is closer to the target reliability level. The statistical parameters of the experimental results are evaluated and the first order-second moment method is used for the calibration. The results vary for these two models and it is necessary to achieve target values of reliability by correcting the models by the resistance factors.