Li－Zhe Jia，Yi－Ming Duan

(School of Civil Engineering，Harbin Institute of Technology，Harbin 150090，China)

1 Introduction

The reliability problem originates from the risk and uncertainty in engineering field.The reliability theory based on probabilistic model is constantly improved and is widely used in engineering design and performance evaluation.Some limits and disadvantages of stochastic model are also disclosed with unceasing improvements and refinements［1－3］.All the classical probabilistic distribution models are idealized mathematics model，and the distributions of uncertain variables are not perfectly matched with these theoretical models.In addition，the statistical analysis of data samples is employed to derive the probabilistic distribution model and critical parameters.The accuracy and objectivity of probabilistic model depend on the quantity and quality of the data.As the data are insufficient，the same group data can match several probabilistic distribution models，and the hypothesis testing can also be passed.Furthermore，the data and information of major building structures are very scarce，and no similar building can be analogical.Especially，the data for major building structures which are suffered from rare events，such as earthquake， tsunami， etc.， become even more shortage.

Numerical simulation can generate a large number of data，but the actual data are also necessary for carrying out simulation and verifying the simulation results.As a result，when the data are unavailable to substantiate the probabilistic model，the results of stochastic model based on subjective assumptions are questionable.In comparison with the probabilistic model in current design code，the non-probabilistic convex set theory proposed by Ben-Haim and Elishakoff［4］are less stringent on data，and only the bounds of uncertain variables are required.As the probabilistic information is unavailable，convex model provides a preferred choice for modeling variables with bounded uncertainty.The convex model-based nonprobabilistic reliability assessmenthas become a popular research field.

The non-probabilistic reliability was firstly proposed by Ben-Haim［5－6］，and the system is reliable if the range of performance fluctuations given uncertain factors remains within an acceptable domain.But，the detailed reliability model is not provided.Furthermore，the non-probabilistic safety factor instead of reliability was derived from the definition of safety factor and interval algorithm by Elishakoff［7］and the structural reliability is approximately reflected with the nonprobabilistic safety factor.However，the application of the safety factor-based method is currently infrequent.A new non-probabilistic reliability model was defined by Ben-Haim［8］based on the robustness of the system to uncertainty.That is，a system is reliable if it can tolerate a large number of uncertainties before failure.This reliability modelisderived with the linear dynamic system and is inapplicable to non-linear systems.

The infinite norm(IN)-based non-probabilistic model was provided by Guo et al.［9］，and the shortest distance from origin to failure surface in the extended space of the normalized variables are used to assess the non-probabilistic reliability.The resistance is not allowed to intersect with the action effect，and the results of the IN model are very conservative.In order to investigate structural reliability when the resistance intersects with the action effect，the ratio of the volume of the safe region to the total volume of the region constructed by basic interval variables was advocated by Wang and Qiu［10－11］as the measure of nonprobabilistic reliability. However， the uncertain variables are assumed to be uniform distribution within the intervals and the idea of probabilistic model is still remained.

The ellipsoidal convex model was also employed to construct the non-probabilistic reliability model［12－13］.The basic variables are defined within the ellipsoidal region，and the mathematical relation of the ellipsoidal equation mustalso be simultaneously met.The ellipsoidal convex model is an appropriate choice when the basic variables meet the ellipsoidal equation.

In this study，the uncertainties of variables were modeled by the envelope bound convex model，and a new non-probabilistic reliability model was proposed after a mathematical transformation of convex variables.Finally，a numericalexample was performed to compare the non-probabilistic reliability model，the infinite norm model and the probabilistic model，and the applicability and rationality of the non-probabilistic reliability model were also verified.

2 Convex Analysis for Non-Probabilistic Reliability Model

2.1 Convex Model and Interval Model

For uncertain variables which vary in a bounded range，both the envelope bound convex model and intervalmodelcan be employed to modelthe uncertainties.Convex model and interval model are two different ways for considering the variable uncertainties based on non-probabilistic methodology.The solution of convex model is usually finished by the extreme optimization method， while the four arithmetic operations and extension principleareapplied to interval model.

The envelope bound convex model in Eq.(1)is one of the simplest convex models and is often used to model the uncertainties of the uncorrelated variables，

where α represents the size of convex model;u(t) indicates the location of convex model;R is a real set.In fact，the interval model［14］in Eq.(2)has the same formula as the envelope bound convex model in Eq.(1)，

where ul(t)and uu(t)are the lower and upper bound of interval model，respectively.The parameter α andof convex model are equivalent to the radius ur(t)= (uu(t)－ul(t))/2 and the center uc(t)=(uu(t)+ ul(t))/2 of interval model，respectively.

2.2 Detailed Procedure of Non-Probabilistic Reliability Model

The limit state equation of two basic variables can be described as

where R and S are the resistance and the action effect，respectively，and vary in a bounded range.The normalization of R and S are

The limit state equation in Eq.(3)is again written as

Following the procedure of FOSM methodology［15］，multiply Eq.(5)by

Let

where η is the shortest distance between origin O'and failure surface in the extended space of the normalized variables as shown in Fig.1(P*is the foot point); cos θδSand cos θδRare the direction cosine of the normal line O'P*.The coordinate of P*in the extended space of the normalized variables is

As(cos θδS)2+(cos θδR)2=1，the nonprobabilistic reliability index η is defined as

The index η is the shortest distance between origin O'and failure surface in the extended space of the normalized variables，and also describes the relative location between the variation range of the basic variables and failure surface(as shown in Fig.1).In this study，the structural reliability is proposed to measure with non-probabilistic way， i.e.， the Euclidean norm minimum.When the failure surface and the normalized variables range are disjoint and the failure surface locates the beneath of the normalized variables range(as shown in Fig.1)，the structure is absolutely reliable.On the contrary，the structure is certainly failure if the failure surface and the normalized variables range are disjoint and the failure surface locates the above of the normalized variables the range(as shown in Fig.2).The structure may be in a safe or failed state when the failure surface intersects with the normalized variables range(as shown in Fig.3).The structure becomes more and more reliable and can tolerate more uncertainties of the basic variables with the increase of the index η.As a result，the non-probabilistic model-based reliability index η can be used to measure structural reliability.However，the non-probabilistic reliability model just indicates structural reliability by the index η，and the failure probability is no longer provided.Similarly with the non-probability index proposed by Ben-Haim［8］，the definition of the index η in this research is also the robustness of the system to uncertainty.

Fig.1 R(Resistance)＞S(Action effect)

The point(S*，R*)is also on the limit state surface，

Fig.2 R(Resistance)＜S(Action effect)

Fig.3 R interferes with S

Consequently，the non-probabilistic reliability index η can be derived from Eqs.(9)－(14)ifˉR，ˉS，αR，αSare given.For multiple basic variables，the analytical procedure of the non-probabilistic reliability index η can be derived from the procedure of two basic variables and theFOSM methodologyofmultiple variables.

3 Reliability Assessment of Steel Beam

A thin-walled steel beam subjects to the dead load (as shown in Fig.4)，and the limit state equation is Z=g(W，f，M)=Wf－M=0.The moment M，sectional resistance moment W and steel strength f are convex variables， and the bounded range is M ∈［102700，157300］，W∈［46.5，62.94］，f∈［2888，4712］，respectively.The non-probabilistic reliability index η is calculated.

Fig.4 Cantilever steel beam

3.1 Non-Probabilistic Reliability Model

According to the bounded range of M，W and f，the location and size parameter of convex model are derived asˉM=130000，αM=27300，ˉW=54.72，αW= 8.22，ˉf=3800，αf=912，respectively.And then，

The direction cosine is calculated by Eqs.(6)－(8)as

The coordinate(W*，f*，M*)is

The limit state equation is again written as

The initial value of W*and f*are assumed to be 54.72 and 3800，respectively.The non-probabilistic reliability index is η=1.41 after three iterations are finished.

3.2 Infinite Norm Model

The procedure of calculating the non-probabilistic reliability index η by IN model［16］is

where δ= {δ1，δ2，…，δn}isthe vectorofthe normalized interval variables;G(δ1，δ2，…，δn)is the limit state function shown by the normalized interval variables.The index η of this example is derived as η= 0.76.

3.3 Non-Probabilistic Reliability Model by Qiu and Wang

The non-probabilistic reliability model proposed by Qiu et al.［10］and Wang et al.［11］as

where Rsetis the reliability probability quantified by set theory;Vsaferegionis the volume of the safe region in the extended space of the normalized variables;Vtotalregionis the total volume.Rsetof the numerical example is derived as 0.9977.

3.4 FOSM(probability)Model

As for the probabilistic reliability model，M，W and f are assumed to be Normal distribution，and the mean and standard deviation are μM=130000，σM= 9100，μW=54.72，σW=2.74，μf=3800，σf=304. The initial value of W*and f*are also 54.72 and 3800，and the probabilistic reliability index is obtained as β=3.80 by FOSM method after three iterations are finished.

The bounded range is again set as M∈［102700，157300］，W∈［46.5，62.94］，f∈［2888，4712］.M，W and f are assumed to be Uniform distribution between the bounds.The initial value of W*and f*are also 54.72 and 3800，and the probabilistic reliability index is obtained as β=2.0 after the Uniform distribution was transformed to Normal distribution.The same to the non-probabilistic model，the variables vary within a bounded range in the Uniform distribution model.Besides，the variables of the Uniform distribution model are assumed uniformly distributed and have to be transformed to Normal distribution.As the number of the assumptions increase，the results become more and more suspicious.

The bounded range of uncertain variables becomes narrower when the information increases.Following，the size parameter of M，W and f are assumed to be 0.1α，0.2α，0.5α，2α，5α，7α，10α，and α is the size parameter ofconvex model.The non-probabilistic reliability index is separately derived as η=12.65，6.33，2.53，0.63，0.25，0.18，0.13.The relation between η and the size parameter of convex model is shown in Fig.5.As can be seen from this example，when the size parameter is 0.2α，and the nonprobabilistic reliability index will be greater than the probabilistic reliability index.That is，the probabilistic model is more conservative than the non-probabilistic model.

Fig.5 η vs.size parameter

4 Comparison of Different Reliability Model

4.1 Different Non-Probabilistic Reliability Model

The non-probabilistic index of the IN model is the shortest distance from origin to failure surface in the extended space of the normalized variables，and the distance is measured by infinite norm.On the contrary，the distance is represented by Euclidean norm in the non-probabilistic index proposed in this research.The probabilistic reliability index β，which is widely used by engineers，is just described by Euclidean norm，i.e.，the distance in common sense.The computation method ofthe IN model， including definitional procedure， transformational procedure， optimal procedure and affineprocedure，is also new for engineers.However，the computation procedure of the non-probabilistic model is similar with the FOSM method，and can be directly applied.Consequently，the non-probabilistic model-based index is easier for engineersand owners to understand and accept compared to the IN index.Furthermore，the index of the IN exists only at one of the points where a hyper radical line，which pass diagonal point of convex polyhedron centered at the origin，intersects with failure surface［17］.The distance between origin and failure surface measured by infinite norm is not the shortest distance，and the relative location between failure surface and the range of the basic variables is not also truly shown.However，the shortest distance between origin and failure surface is accurately presented in the non-probabilistic model.

The variables of the non-probabilistic reliability model of Qiu［10］are assumed to be uniform distribution within the bounds.Any probabilistic distribution information is not required in the proposed nonprobabilistic reliability model.

As for the IN model，the resistance is not allowed to intersect with the action effect，and the structure is absolute reliable.The results of the IN model are very conservative.As shown in the above numerical example，the non-probabilistic reliability index η by IN model is 0.76＜1，and the beam is unreliable.On the contrary， the non-probabilistic reliability model proposed by Qiu et al.［10］and Wang et al.［11］indicated that the likelihood of failure is very small，and the reliability probability is 0.9977.The results of the nonprobabilistic reliability model proposed in this paper (η=1.41)also shown that the reliability of this beam is rather high，and the failure surface is almost tangential with the normalized variables range.

The non-probabilistic-based model also needs to determine of the acceptable level of the nonprobabilistic reliability index.Similarly with the determination of the allowable value of the probabilistic reliability index，the economic level of the country，the technical capability of engineers and the needs of society and owners should be comprehensively considered for determining the acceptable level of the non-probabilistic reliability index.

4.2 Probabilistic Model and Non-Probabilistic Reliability Model

The probability density function is an integral part of probabilistic model，while only the range of uncertain variables is required in the non-probabilistic reliability model. The equivalent normalize transformations is certainly unnecessary for the nonprobabilistic reliability model，and the computation procedure is similar with the FOSM method.As for the probabilistic reliability model，both the reliability index β and the failure probability are equivalent and can be the indicators of structural reliability.However，the structural reliability is measured by non-probabilistic way in the non-probabilistic reliability model，and the failure probability is no longer provided.When the size parameter is small enough and the non-probabilistic reliability index η will be greater than the probabilistic reliability index β.That is，the probabilistic model is more conservative than the non-probabilistic model，and this conclusion is also verified in the numerical example(as shown in Fig.5).The range of the uncertain variables become more and more narrow with the increase ofthe data and information， and eventually will be a constant parameter.Consequently，the results of the non-probabilistic model will become more and more accurate.As a result，the probabilistic and non-probabilistic reliability model provides two different ways for quantifying risk，and the available data and information decide which model should be adopted.

5 Conclusions

A new non-probabilistic reliability model based on the robustness of system to uncertainty was proposed.The rationality of the new non-probabilistic reliability model includes that the resistance is allowed to couple with the action effect，and the distribution model of variables is no longer required.Also，the idea and computation procedure of this new non-probabilistic reliability model are easier for engineers and owners to understand and accept compared to the IN model.Additionally，the probabilistic and non-probabilistic reliability modelprovides two differentways for quantifying risk，and the available data and information decide which model should be adopted.

The structural reliability can be effectively measured by the non-probabilistic model proposed in this research when the data and information are scarcely.The non-probabilistic reliability model is an instructive complement to theprobabilistic model.Simultaneously，it is still a top issue to be addressed about how to determine of the acceptable level of the non-probabilistic reliability index.

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