Time-Dependent Reliability Modeling and Analysis Method for Mechanics Based on Convex Process

The objective of the present study is to evaluate the time-dependent reliability for dynamic mechanics with insufficient timevarying uncertainty information. In this paper, the nonprobabilistic convex process model, which contains autocorrelation and cross-correlation, is firstly employed for the quantitative assessment of the time-variant uncertainty in structural performance characteristics. By combination of the set-theorymethod and the regularization treatment, the time-varying properties of structural limit state are determined and a standard convex processwith autocorrelation for describing the limit state is formulated. By virtue of the classical first-passage method in random process theory, a new nonprobabilistic measure index of time-dependent reliability is proposed and its solution strategy ismathematically conducted. Furthermore, theMonte-Carlo simulationmethod is also discussed to illustrate the feasibility and accuracy of the developed approach. Three engineering cases clearly demonstrate that the proposed method may provide a reasonable and more efficient way to estimate structural safety than Monte-Carlo simulations throughout a product life-cycle.


Introduction
Structural reliability assessment aims at computing the probability/possibility of failure occurrence for a mechanical system with reference to a specific failure criterion by accounting for uncertainties arising in the model description or the environment [1].Essentially, the reliability measurement represents safety level in industry practice and hence the capability to efficiently perform reliability analysis is of vital importance in practical engineering applications [2,3].Currently, two main categories of reliability analysis are, respectively, the time-independent (static) reliability analysis and the timedependent reliability analysis [4].Over the past few decades, many efforts have been focused on static reliability research methodologies including probabilistic [5,6], nonprobabilistic [7,8], and hybrid [9,10] models.Therefore, the timeindependent reliability theory has made great progress in the reliability estimate of all kinds of industrial systems and becomes the most universal theory when dealing with uncertainties.
However, in view of the comprehensive reasons of material property degeneration, varying environmental conditions, and dynamic load processes, the uncertain structures in practical engineering still exhibit a distinct time-varying effect [11].Although the approaches based on static reliability theory have been widely used to estimate the structural safety recently, how to ensure high reliability level during a product life-cycle is still a big challenge in engineering applications.To tackle the time-dependency issues, Lots of research results have been published in time-dependent reliability analysis in recent years.They include the extreme value distribution method [12,13], the first-passage method [1,14], the Markov chain method [15], and the Monte-Carlo simulation method [16].Among them, the first-passage method based on Rice's formula [17] is recognized as the most popular method in current literature of reliability analysis research.It concentrates on the first time when the performance function exceeds the upper bound or falls below the lower bound of the given safety threshold, which is commonly by virtue of the calculation of the "outcrossing rate" to quantitatively evaluate the probability of failure.Since the development of Rice's formula, amounts of improvements have been made.For example, Vanmarcke proposed a commonly used improved formula, accounting for the dependence between the crossing events and the time that the process spends above the barrier in application to normal stationary random process model 2 Mathematical Problems in Engineering [18]; Madsen and Krenk developed an integral equation method for solving the first-passage problems [19]; a timedependent reliability analysis method with joint upcrossing rates, inspired by [19], was further developed by Hu and Du [20] for more general cases of the limit-state functions that involve time, random variables, and stochastic processes; by combination of the ideas of outcrossing and system reliability, Wang et al. [13] presented an improved subset simulation with splitting approach by partitioning the original high dimensional random process into a series of correlated, short duration, low dimensional random processes; several improved formulations for calculating the outcrossing rate based on the Poisson assumption were, respectively, extended by Schall et al. [21], Engelund et al. [22], Streicher and Rackwitz [23], and so forth.Recently, additional correlational researches have been also suggested by [24,25].
Compared to static reliability analysis, the research on time-dependent reliability is still in its preliminary stage, and the following features contribute to the sources of difficulty: (1) from the analytical point of view, very few models or analysis methods have been presented in the past for the evaluation of time-dependent reliability considering the general case of non-Gaussianity, nonstationarity, and nonlinear dependency; (2) from the perspective of simulation, extremely high computational cost is needed to guarantee reasonable numerical results of time-dependent reliability, which greatly restrict the application in complex structural systems.In recent developments, the work of Sudret et al. [1,26], where an analytical outcrossing rate utilizing the PHI2 method is derived, may partly overcome the abovementioned insufficiencies.
As the literature survey reveals, most of the existing studies focus on the random process models when performing the time-dependent reliability analysis while ignoring the existence of nonprobabilistic time-varying uncertainties.Furthermore, the assessment of the time-dependent reliability by random process theory must require knowledge of probabilistic descriptions for all time-varying uncertainties, which are typically determined by sufficient experimental samples.Unfortunately, for practical engineering problems, experimental samples are not always available or re sometimes very expensive to obtain so that one cannot directly establish the precise analytical model.Giving subjective assumptions for description of the uncertainty characteristics is likely to bring about a serious error of the time-dependent reliability results.In view of the above reasons, it is quite necessary and urgent to carry out time-dependent reliability research on nonprobabilistic modeling methods.Jiang et al. [27] proposed a new theory for time-varying uncertainty, namely, the "non-probabilistic convex process model" to tackle the problems of safety estimate when lacking relevant uncertainty information.However, the solution of the timedependent reliability must rely on the Monte-Carlo simulations rather than constructing an analytical model/index.Additionally, the work in [27] mainly concentrated on the case of stationary convex process; the more common case of non-stationary process was not discussed.
Generally speaking, compared with static reliability analysis, fewer studies on time-dependent reliability analysis are performed at present and its theoretical foundation is still relatively immature.In this paper, inspired by the work in [26,27], we develop a new time-dependent reliability analysis approach based on the nonprobabilistic convex process model, in which a new measure index of time-dependent reliability is established and its solution procedure is deduced mathematically.The presented approach can effectively deal with the more general case containing nonstationarity and nonlinear dependency that is of particular importance in solving practical engineering problems.
The paper is organized as follows.Section 2 provides a brief review of general concept of nonprobabilistic convex process model and its relevant mathematical basis.In Section 3, combined with the foregoing convex process model and the set-theory method, the time-varying uncertainty of structural limit state is quantified.Section 4 details the presented time-dependent reliability method, and then we introduce the Monte-Carlo simulations in Section 5.The proposed methodology is demonstrated with three case studies in Section 6 and then followed by conclusions in Section 7.

Notation and Classification of Nonprobabilistic Convex Process Model
Enlightened by [27], the nonprobabilistic convex process model is introduced in this section to quantify the timevarying uncertain parameters in dynamic mechanics when facing the case of limited sample information.Here, the uncertainty of structural parameters at any time is depicted with a bounded closed interval, and the correlation between variables in different instant time points is reflected by defining a corresponding correlation function.If we discretize the contiguous convex process into a time series, the feasible domain of all interval variables belongs to a convex set.For details, see the following definitions.
Definition 1.Consider a convex process (); () and () describe the upper and lower bound functions of (), and hence the mean value function   () and the radius function   () are, respectively, given by ( For convenience, we also define the variance function   () as It is apparently indicated that the properties of uncertain variables at any time can be quantified mathematically from Suppose that ( 1 ) and ( 2 ) are two interval variables originating from the convex process () at times  1 and  2 .In view of the existence of the correlation, a rotary ellipse model is constructed to restrict the value range of ( 1 ) and ( 2 ), as shown in Figure 1(a).By regularization treatment (from  space to  space), the standard model is then formed, as illustrated in Figure 1(b).Definition 2. If () is a convex process, for any times  1 and  2 , the autocovariance function of interval variables ( 1 ) and ( 2 ) is defined as where   either stands for the major axis  1 of the ellipse in  space if the slope of the major axis is positive or represents the minor axis  2 if the slope of the major axis is negative.For details, see Figure 2.
Definition 3.With regard to the convex process (), the autocorrelation coefficient function of ( 1 ) and ( 2 ) is expressed as where  3.Among all the ellipses, the one with   ( 1 ,  2 ) = 0 has a maximal area, which implies the largest scattering degree and hence a minimum correlativity of variables ( 1 ) and ( 2 ).Moreover, a larger |  ( 1 ,  2 )| means a stronger linear correlation between ( 1 ) and ( 2 ).Once |  ( 1 ,  2 )| = 1, the ellipse model will be replaced by a straight line that indicates a complete linearity.
As mentioned above, the characteristics of one timevarying uncertain parameter can be commendably embodied.However, with respect to complicated structures, more common case is that we must solve the problem of multidimensional time-varying uncertainty.The cross-correlation between two convex processes should be considered as well.Definition 4. Suppose two convex processes () and (), for any times  1 and  2 ; the cross-correlation coefficient function of ( 1 ) and ( 2 ) is established as where Cov  ( 1 ,  2 ) means the cross-covariance function and   * represents the major/minor axis of the ellipse in  space derived from ( 1 ) and ( 2 ) (similarly to the definition in (3)).
According to the above definitions, the characteristic parameters of nonprobabilistic convex process model are available.In the next section, associated with the set-theory method, the convex process model utilized to describe the time-varying uncertainty of structural limit state is established.

Time-Varying Uncertainty Quantification Analysis Corresponding to the Limit State
With regard to any engineering problems, it is extremely significant to determine the performance of structural limit state.Once time-varying uncertainties originating from inherent properties or external environmental conditions are considered, the limit state should have time-variant uncertainty as well.In fact, if we define the time-varying structural parameters as convex processes, the limit-state function can also be enclosed by a model of convex process.The detailed procedure of construction of the convex process model for structural limit state is mathematically discussed in the following statements.
In addition, taking into account the comprehensive effect of the autocorrelation between   ( 1 ) and   ( 2 ) as well as the cross-correlation between   ( 1 ) and   ( 2 ), the autocovariance function corresponding to ( 1 ) and ( 2 ) is found to be where   ( 1 ,  2 ) is the autocorrelation coefficient function of   ( 1 ) and   ( 2 ) and      ( 1 ,  2 ) is the cross-covariance function of   ( 1 ) and   ( 2 ).
In terms of the definitions in (5), we arrive at It is instructive to discuss the case of ((), ()) = ()− (), in which the strength () and the stress () are both enclosed by convex processes; we get In accordance with the above equations, the time-varying uncertainty of limit-state () is explicitly embodied by a nonprobabilistic convex process model.That is to say, for any times  1 and  2 , once the time-variant uncertainties existing in structural parameters are known, the ellipse utilized to restrict the feasible domain between ( 1 ) and ( 2 ) can be affirmed exclusively.
Apparently, it should be indicated that if the limitstate function is nonlinear, several linearization techniques, such as Taylor's series expansion and interval perturbation approach, may make effective contributions to help us construct its convex process model.

Time-Dependent Reliability Measure Index and Its Solving Process
4.1.The Classical First-Passage Method in Random Process Theory.For engineering problems, we are sometimes more interested in the calculation of time-varying probability/possibility of reliability because it provides us with the likelihood of a product performing its intended function over its service time.From the perspective of randomness, the time-dependent probability of failure/safety during a time interval [0, ] is computed by or where X() is a vector of random processes and Pr{} stands for the probability of event .Generally, it is extremely difficult to obtain the exact solution of   () or   ().At present, the most common approach to approximately solve time-dependent reliability problems in a rigorous way is the so-called first-passage approach (represented schematically in Figure 4).
The basic idea in first-passage approach is that crossing from the nonfailure into the failure domain at each instant time may be considered as being independent of one another.Denote by  + (0, ) the number of upcrossings of zero-value by the compound process (, X()) from safe domain to the failure domain within [0, ], and hence the probability of failure also reads Then, the following upper bound on   () is available; that is, First-passage outcrossing where   (0) is the probability of failure when time is fixed as zero and ]() is the instantaneous outcrossing rate at time .
The specific expression of ]() is evaluated as Introducing the finite-difference concept, rewrite (15) in the form where Δ has to be selected properly (under the sufficiently small level).

Time-Dependent Reliability Measurement Based on Nonprobabilistic Convex Process Model.
Although the firstpassage approach tends to provide a rigorous mathematical derivation of the time-dependent reliability, it is actually difficult to evaluate the outcrossing rate for the general random processes.Only in few cases, an analytical outcrossing rate is available.Furthermore, under the case of limited sample data of time-varying uncertainty, the analysis based on random process theory is inapplicable.Therefore, in this section, we apply the idea of the first-passage method into nonprobabilistic convex process model and further establish a new measure index for evaluating time-dependent reliability with insufficient information of uncertainty.The details are as follows: (i) Acquire convex process corresponding to the limitstate () (by reference to the corresponding Sections 2 and 3).
(iv) Redefine the outcrossing rate ](Δ) as where Pos{  } stands for the possibility of event   , which is regarded as the ratio of the interference area to the whole elliptic area ( = 1, 2, . . ., ); that is, (v) Determine the time-dependent reliability measure index; namely, (20)

Solution Strategy of Time-Dependent Reliability Measure
Index.Obviously, the key point for determining the proposed time-dependent reliability measure index is just the calculation of Pos{  } (for practical problems of ( 2 / √ 2)  () <   () <   (), as shown in Figure 5(a)).
As mentioned above, the regularization methodology is firstly applied (as shown in Figure 5(b)), and the event   is equivalent to For the sake of convenience, one more coordinate transformation is then carried out, and a unit circular domain is eventually obtained (as shown in Figure 6).Equation ( 21) leads to From Figure 6, it becomes apparent that the shaded area   indicates the interference area   interference , and the total area   total always equals .Hence, the analytical expression of   is of paramount importance, and several following steps should be executed.
Step 1 (solution of the coordinates of points  and ).Let us consider the simultaneous equations as By elimination method, a quadratic equation is further deduced; that is, Thus, the coordinates of points  and  on  2 -axis are, respectively, Mathematical Problems in Engineering Substituting ( 25) and ( 26) into (23), the coordinates of points  and  on  1 -axis, namely,  1  and  1  , can be easily obtained.
Step 2 (solution of the coordinates of points  and ).Taking into account the simultaneous equations we arrive at Therefore, the coordinates of points  and  on  2 -axis are evaluated as Substituting ( 29) and (30) into (27), the coordinates of points  and  on  1 -axis, namely,  1  and  1  , can be easily determined.
Step 3 (solution of the coordinates of point ).Since  is actually the point of intersection between  1 and  2 , the coordinate calculation is satisfied: Then we have By virtue of the above three steps, the geometric information of the characteristic points (from  to ) can be obtained mathematically, and the coordinates of these points are then used to deduce the analytical expression of   interference (  ).In fact, the physical mean for definitions of the characteristic points lies in that the possibility of the cross failure during a small time interval can be eventually embodied by a form of geometric domain, which is determined by the points.
Step 4 (solution of the fan-shaped area   0  ).The central angel ∠ is firstly computed by The area   0  equals where  represents the radius of the unit circular domain.
Step 5 (solution of the triangular areas  Δ and  Δ ).As previously noted, the coordinates of points , , and  have been available.Thus, the triangular area  Δ is derived from Similarly,  Δ is Step 6 (solution of the bow-shaped area  •  ).It can be found that the bow-shaped area  •  is denoted by the difference with the fan-shaped area   0  and the triangular area  Δ ; namely, where  is the chord length and , as the distance from origin to line  1 , equals   (Δ)/  (Δ).
Sample generation: Calculation of g(jΔt) by x(jΔt): {x 1 (jΔt), x 2 (jΔt), , x n (jΔt)}  Step 7 (solution of Pos{  }).Consider We should calculate Pos{  } successively ( = 1, 2, . . ., ) based on the above steps.By combination with the analysis stated in Section 4.2, an integral procedure for construction of the time-dependent reliability measure index can be eventually achieved.For ease of understanding, one flowchart is further added (see Figure 7 for details).

Monte-Carlo Simulation Method
For comparison's purpose, this paper also provides a Monte-Carlo simulation method to compute structural timedependent reliability.The detailed analysis procedure is expounded below (as is seen in Figure 8): (i) Initialize the numbers of generated samples and failed samples as  generated = 0,  failed = 0; define the total number of samples as  total ; the cyclic counting index  is set to zero.
( By means of the proposed methodology of Monte-Carlo simulations, the numerical results of time-dependent reliability can be obtained in principle.However, some important issues remain unsolved in practical applications: (1) there is high complexity in sample generation for all convex processes, which is mainly embodied by the construction of multidimensional ellipsoidal convex models (Ω 1 , Ω 2 , . . ., Ω  ) under the case of small time increment Δ.(2) The above analysis is not able to demonstrate the cross-correlation between two convex processes.(3) The given assumption of "3 rule" may result in an extremely large error when encountering the case of the combination of various uncertainty characteristics.( 4) Enormous computational costs have to be confronted when solving complex engineering problems.Therefore, compared with the Monte-Carlo simulations, the presented method based on nonprobabilistic convex process model may show superiority to some extent when dealing with the problems of time-dependent reliability evaluation.

Numerical Examples
In this section, three engineering examples are investigated.Among them, the first two examples of a cantilever beam structure and a ten-bar truss structure are further analyzed by Monte-Carlo simulations, and hence the numerical results can be regarded as a reference to effectively demonstrate the validity and feasibility of the presented methodology.That is to say, the accuracy of the proposed method is verified by the simulation techniques, and the deviation of the reliability results may quantify a specific precision level.
Additionally, the safety estimation of a complicated propeller structure in the last example can better illustrate the advantage and capability of the developed time-dependent reliability method when tackling reliability issues of large complex structures.

A Cantilever Beam Structure.
A cantilever beam structure modified from the numerical example in [28] is considered, as shown in Figure 9. Three time-varying external forces  1 (),  2 (), and  3 () are applied to the beam, and the maximum moment on the constraint surface at the origin should be less than an allowable value  criteria ().Thus, the following limitstate function can be created by where  1 = 1 m,  2 = 2 m, and  = 5 m and  in this problem is defined as 10 years.
The time-dependent reliability results   () obtained by the nonprobabilistic convex process model are given in Table 2 and Figure 10 (Δ = 0.05 years).It can be found that values of   () decrease remarkably when either  or  Table 1: Time-varying uncertainty characteristics of the cantilever beam structure.
Mean value functions and radius functions of convex processes:  1 (),  2 (),  3 (),  criteria () Table 2: Reliability results based on the nonprobabilistic convex process model.increases, as expected.This indicates that higher dispersion or weaker correlation leads to lower structural safety.For example, the cantilever beam is absolutely safe when  and  are both equal to 1, while   () only reaches 0.6972 under the case of  = 2 and  = 5.
To better analyze the accuracy of the time-dependent reliability results, the Monte-Carlo simulation method is also used to deal with a severe case of  = 2 (100000 samples).The comparative results are illustrated in Figure 11.As shown in Figure 11,   () calculated by our method is coincident with the results derived from the numerical simulations.However, there are three points that should be noted: (1) The results based on convex process model are more conservative due to the fact that less information on time-varying uncertainty is needed.(2) Associated with the increasing values of , the deviation of the reliability results   () is widening (from 2.90% to 5.92%).( 3) The accuracy of the reliability from Monte-Carlo simulations mainly relies on the number of samples which implies that the situation of intensive computation and consuming time may have to be confronted.

A Ten-Bar Truss Structure.
A well-known ten-bar truss structure modified from the numerical example in [15] is investigated, as depicted in Figure 12.Young's modulus  = 68948 MPa, and the density  = 7.85 × 10 −9 /mm 3 .The length  of the horizontal and vertical bars is 9144 mm, and the area of each bar equals 4000 mm  Mean value functions and radius functions of convex processes:  1 (),  2 (),  3 (),  ,criteria (),  ,criteria () Taking into account the cross-correlation between  1 () and  2 () where  = 10 years.
It is not difficult to understand that member 4 and member 8 are most dangerous, respectively, under the tensile situation and compressive situation.According to the definitions in (41) and (42), the time-dependent reliability results   () obtained by the nonprobabilistic convex process For the purpose of verification and comparison, the Monte-Carlo simulation method is utilized again to analyze the case of   1  2 = 0 (in disregard of cross-correlation).The time-dependent reliability results are illustrated in Figures 15  and 16.Similarly to conditions on the first example, the timedependent reliability results calculated by convex process model are consistent with the results computed by the Monte-Carlo simulation method in qualitative analysis but lower to some extent from quantitative perspective.Specifically, when  = 2, the maximal deviations of reliability results are, respectively, 13.06% for member 4 and 26.17% for member 8.

A Marine Propeller
Structure.The high-speed rotating propeller structure, as a vital component of power plant, must reach high standard of safety during its whole lifetime   to guarantee normal operations of marine system.In actual working conditions, affected by changeable environment of water flow, the forces experienced on propeller are always complicated and are uncertain at different times.Therefore, the time-dependent reliability analysis of this structure is of great significance.Figure 17 shows a specific type of propeller, which is composed of four blades and one support block, and its material parameters are listed in Table 6.Three types of time-varying forces are applied to the surface of each blade, namely, the propulsive force  1 (), the centrifugal force  2 (), and the shear force  3 ().Hence, the limit-state function is expressed as  Assume that dynamic forces  1 (),  2 (), and  3 () as well as allowable strength  criteria () are all convex processes, where autocorrelation and cross-correlation are both considered.The detailed information of time-varying uncertainty is summarized in Table 7 where  can be valued by 0.1, 0.2, . . ., 1.
The present method is applied to the above engineering problem, and all the analysis results are given in Table 8 and Figure 18 (Δ = 0.05 years).It can be seen that ( 1) there is no need to worry about failure when  = 0.1 or 0.2 (  () = 1).
(2) As  increases, the time-dependent reliability may show a decreasing trend (the minimum   () = 0.7538 under the case of  = 1), and its variation is essentially linear at the beginning and then becomes nonlinear to a certain degree.(1) The characteristics of the time-varying uncertain parameters generally exert a great influence on structural safety.On the one hand, the increasing dispersion and decreasing autocorrelation may result in a severe situation of reduced time-dependent reliability (mainly consulted by Section 6.1); on the other hand, the role of cross-correlation between time-varying uncertain parameters makes analysis more complicated (referred by Section 6.2). (2) Because less assumptions of uncertainty are needed, more conservative reliability results given by present method appear than those from numerical simulations.However, it should be emphasized that the structural reliability is closely related to the uncertain parameters, and hence subjective assumptions may yield unreliable results, especially when dealing with the engineering cases of limited samples.(3) Directed at simple problems of dynamics, the Monte-Carlo simulation method, as an optional way, can evaluate structural reliability by means of sufficient samples and cumulative operations (such as Sections 6.1 and 6.2) but is powerless when tackling mechanical problems, which contain multidimensional uncertainties, cross-correlation, largescale configurations, or complex boundary conditions (as stated in Section 6.3 in this study).
In summary, the numerical examples demonstrate that the time-dependent reliability analysis based on nonprobabilistic convex process model has high efficiency and simultaneously an acceptable analysis precision.More importantly, we can provide a feasible and reasonable way to mathematically evaluate the dynamical safety for complex engineering problems.

Conclusions
With the rapid technological advance, the reliability analysis considering time-varying effect has attracted more and more concerns and discussions.Currently, most of the approaches for performing time-dependent reliability assessment are always based upon the random process model, where the distributions of time-varying uncertain parameters should be determined by a substantial number of samples, which, however, are not always available or sometimes very costly for practical problems.Thus some assumptions on distribution characteristics have to be made in many dynamical cases when using the probability model.Nevertheless, unjustified assumptions may give rise to misleading results unexpectedly.
In view of the abovementioned facts, this paper describes the time-varying uncertainty with the model of nonprobabilistic convex process.In this convex process model, the uncertain variables at any time are expressed as intervals and the corresponding autocovariance function and autocorrelation coefficient function are established to characterize the relationship between variables at different times.Referred by the definitions in random process theory, the cross-correlation between different two convex processes is also considered.Then, by using the set-theory method and the regularization technique, the time-varying limit-state function is transformed and quantified by a standard convex process model with autocorrelation.Enlightened by the ideas of first-passage method and static reliability analysis, a new nonprobabilistic measurement of time-dependent reliability is proposed, and its analytical expression in linear case is conducted mathematically.Additionally, as a means of verification and comparison, the Monte-Carlo simulation method is also presented and applied into the solution of numerical examples.Analytical results indicate that the present method can be ensured to be more applicable and efficient when estimating structural safety of complex engineering problems.
Indeed, the present time-dependent reliability analysis technique based on the nonprobabilistic convex process model can be regarded as a beneficial supplement to the current reliability theory of random processes.On the basis of the reliability analysis, the proposed safety measurement index can be also applied to the fields of time-varying structural design optimization.

Figure 1 :
Figure 1: Quantitative models between interval variables at any two times in the convex process: (a) the original model; (b) the normalized model.

Figure 2 :
Figure 2: Definitions of autocovariance function between interval variables at any two times in the convex process: (a) the positive autocorrelation; (b) the negative autocorrelation.

Figure 3 :
Figure 3: Typical geometric shapes of ellipses for different correlation coefficients.

Figure 4 :
Figure 4: Schematic diagram for the first-passage approach.

Figure 5 :
Figure 5: Uncertainty domains with regard to the limit-state function: (a) the original domain; (b) the regularized domain.

Figure 6 :
Figure 6: Schematic diagram for the solution of time-dependent reliability measure index.

Figure 7 :
Figure 7: Flowchart of the time-dependent reliability analysis based on nonprobabilistic convex process model.

1 N 1 N 1 .
failed = N failed + generated ≤ N total N generated N generated = N generated +

Figure 8 :
Figure 8: Flowchart of the time-dependent reliability analysis based on the Monte-Carlo simulation method.

Figure 10 :
Figure 10: Time-dependent reliability results versus  and .

Figure 11 :
Figure 11: Comparisons of the reliability results obtained by the convex process model and the Monte-Carlo based model.

Figure 15 :Figure 16 :
Figure 15: Comparisons of the reliability results obtained by the convex process model and the Monte-Carlo based model (member 4).
1 and   2 denote the variance functions of the standard interval variables  1 and  2 (  1 =   2 = 1).( 1 ,  2 ) is a dimensionless quantity and its magnitude represents the linear correlation of ( 1 ) and ( 2 ).It is obvious that|  ( 1 ,  2 )| ≤ 1 and   (, ) = 1.Several specific ellipse models with different values of   ( 1 ,  2 ) are clearly illustrated in Figure ) if not, reset  to zero and go back to (iii).(v)The structural failure possibility is approximately computed as   () =  failed / total ; furthermore, the time-dependent reliability is obviously depicted as   () = ( total −  failed )/ total .

Table 3 :
Time-varying uncertainty characteristics of the ten-bar truss structure.

Table 4 :
Reliability results based on the nonprobabilistic convex process model (member 4).

Table 5 :
Reliability results based on the nonprobabilistic convex process model (member 8).It is remarkable that (1) the results of   () continue to decline as the increase of .(2) If   1  2 is negative, stronger cross-correlation may lead to a higher level of structural safety; if   1  2 is positive, however, the truss will be more dangerous with an increasing   1  2 .(3)   1  2 has a minor effect on reliability results for member 4, while it plays an important role in safety estimation for member 8.

Table 6 :
Material parameters of the marine propeller structure.Kg/m3where  max () denotes the maximum von Mises stress of the propeller structure,  criteria () is allowable strength, and  = 10 years.Considering that the propeller belongs to a complicated three-dimensional structure, the dynamic finite element model, containing 44888 elements and 31085 nodes, is set up to estimate the structural safety.

Table 8 :
Reliability results based on the nonprobabilistic convex process model.Discussions on the Computational Results.Synthesizing the computational results of the above three numerical examples, the following points can be inherited: