Research Article Time-Dependent Reliability Modeling and...

Research ArticleTime-Dependent Reliability Modeling and Analysis Method forMechanics Based on Convex Process

Lei Wang Xiaojun Wang Ruixing Wang and Xiao Chen

Institute of Solid Mechanics Beihang University Beijing 100191 China

Correspondence should be addressed to Xiaojun Wang xjwangbuaaeducn

Received 1 August 2014 Revised 30 March 2015 Accepted 30 March 2015

Academic Editor Nenad Mladenovic

Copyright copy 2015 Lei Wang et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

Structural reliability assessment aims at computing the prob-abilitypossibility of failure occurrence for a mechanical sys-temwith reference to a specific failure criterion by accountingfor uncertainties arising in the model description or the envi-ronment [1] Essentially the reliability measurement repre-sents safety level in industry practice and hence the capabilityto efficiently perform reliability analysis is of vital importancein practical engineering applications [2 3] Currently twomain categories of reliability analysis are respectively thetime-independent (static) reliability analysis and the time-dependent reliability analysis [4] Over the past few decadesmany efforts have been focused on static reliability researchmethodologies including probabilistic [5 6] nonprobabilistic[7 8] and hybrid [9 10] models Therefore the time-independent reliability theory has made great progress inthe reliability estimate of all kinds of industrial systemsand becomes the most universal theory when dealing withuncertainties

However in view of the comprehensive reasons of mate-rial property degeneration varying environmental condi-tions and dynamic load processes the uncertain structures

in practical engineering still exhibit a distinct time-varyingeffect [11] Although the approaches based on static reliabilitytheory have been widely used to estimate the structural safetyrecently how to ensure high reliability level during a productlife-cycle is still a big challenge in engineering applicationsTo tackle the time-dependency issues Lots of research resultshave been published in time-dependent reliability analysisin recent years They include the extreme value distribu-tion method [12 13] the first-passage method [1 14] theMarkov chain method [15] and the Monte-Carlo simulationmethod [16] Among them the first-passage method basedon Ricersquos formula [17] is recognized as the most popularmethod in current literature of reliability analysis research Itconcentrates on the first timewhen the performance functionexceeds the upper bound or falls below the lower bound of thegiven safety threshold which is commonly by virtue of thecalculation of the ldquooutcrossing raterdquo to quantitatively evaluatethe probability of failure Since the development of Ricersquosformula amounts of improvements have been made Forexample Vanmarcke proposed a commonly used improvedformula accounting for the dependence between the crossingevents and the time that the process spends above the barrierin application to normal stationary random process model

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 914893 16 pageshttpdxdoiorg1011552015914893

2 Mathematical Problems in Engineering

[18] Madsen and Krenk developed an integral equationmethod for solving the first-passage problems [19] a time-dependent reliability analysis method with joint upcrossingrates inspired by [19] was further developed by Hu and Du[20] for more general cases of the limit-state functions thatinvolve time random variables and stochastic processes bycombination of the ideas of outcrossing and system reliabilityWang et al [13] presented an improved subset simulationwith splitting approach by partitioning the original highdimensional random process into a series of correlatedshort duration low dimensional random processes severalimproved formulations for calculating the outcrossing ratebased on the Poisson assumptionwere respectively extendedby Schall et al [21] Engelund et al [22] Streicher andRackwitz [23] and so forth Recently additional correlationalresearches have been also suggested by [24 25]

Compared to static reliability analysis the research ontime-dependent reliability is still in its preliminary stage andthe following features contribute to the sources of difficulty(1) from the analytical point of view very few modelsor analysis methods have been presented in the past forthe evaluation of time-dependent reliability considering thegeneral case of non-Gaussianity nonstationarity and non-linear dependency (2) from the perspective of simulationextremely high computational cost is needed to guaranteereasonable numerical results of time-dependent reliabilitywhich greatly restrict the application in complex structuralsystems In recent developments the work of Sudret et al[1 26] where an analytical outcrossing rate utilizing the PHI2method is derived may partly overcome the abovementionedinsufficiencies

As the literature survey reveals most of the existingstudies focus on the random process models when perform-ing the time-dependent reliability analysis while ignoringthe existence of nonprobabilistic time-varying uncertaintiesFurthermore the assessment of the time-dependent relia-bility by random process theory must require knowledge ofprobabilistic descriptions for all time-varying uncertaintieswhich are typically determined by sufficient experimentalsamples Unfortunately for practical engineering problemsexperimental samples are not always available or re some-times very expensive to obtain so that one cannot directlyestablish the precise analytical model Giving subjectiveassumptions for description of the uncertainty characteristicsis likely to bring about a serious error of the time-dependentreliability results In view of the above reasons it is quitenecessary and urgent to carry out time-dependent reliabilityresearch on nonprobabilistic modeling methods Jiang et al[27] proposed a new theory for time-varying uncertaintynamely the ldquonon-probabilistic convex process modelrdquo totackle the problems of safety estimate when lacking relevantuncertainty information However the solution of the time-dependent reliability must rely on the Monte-Carlo simu-lations rather than constructing an analytical modelindexAdditionally the work in [27] mainly concentrated on thecase of stationary convex process the more common case ofnon-stationary process was not discussed

Generally speaking compared with static reliability anal-ysis fewer studies on time-dependent reliability analysis are

performed at present and its theoretical foundation is stillrelatively immature In this paper inspired by the work in[26 27] we develop a new time-dependent reliability anal-ysis approach based on the nonprobabilistic convex processmodel in which a new measure index of time-dependentreliability is established and its solution procedure is deducedmathematically The presented approach can effectively dealwith the more general case containing nonstationarity andnonlinear dependency that is of particular importance insolving practical engineering problems

The paper is organized as follows Section 2 providesa brief review of general concept of nonprobabilistic con-vex process model and its relevant mathematical basis InSection 3 combinedwith the foregoing convex processmodeland the set-theory method the time-varying uncertaintyof structural limit state is quantified Section 4 details thepresented time-dependent reliability method and then weintroduce the Monte-Carlo simulations in Section 5 Theproposed methodology is demonstrated with three casestudies in Section 6 and then followed by conclusions inSection 7

2 Notation and Classification ofNonprobabilistic Convex Process Model

Enlightened by [27] the nonprobabilistic convex processmodel is introduced in this section to quantify the time-varying uncertain parameters in dynamic mechanics whenfacing the case of limited sample information Here theuncertainty of structural parameters at any time is depictedwith a bounded closed interval and the correlation betweenvariables in different instant time points is reflected by defin-ing a corresponding correlation function If we discretize thecontiguous convex process into a time series the feasibledomain of all interval variables belongs to a convex set Fordetails see the following definitions

Definition 1 Consider a convex process 119883(119905) 119883(119905) and 119883(119905)describe the upper and lower bound functions of 119883(119905) andhence the mean value function119883119888

(119905) and the radius function119883

119903

(119905) are respectively given by

119883119888

(119905) =

119883 (119905) + 119883 (119905)

2

119883119903

(119905) =

119883 (119905) minus 119883 (119905)

2

(1)

For convenience we also define the variance function119863

119883(119905) as

119863119883(119905) = (119883

119903

(119905))2= (

119883 (119905) minus 119883 (119905)

2)

2

(2)

It is apparently indicated that the properties of uncertainvariables at any time can be quantified mathematically from

Mathematical Problems in Engineering 3

0

(a) (b)

0

02

2

X(t2)

X(t2)

Xc(t2)

X(t2)

X(t1)X(t1)Xc(t1)X(t1)

Xr(t1)

Xr(t2)

U1

U2

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

the above equations The correlation between any two vari-ables in different time points is determined by the autoco-variance function or the correlation coefficient function asthe following statements

Suppose that 119883(1199051) and 119883(1199052) are two interval variablesoriginating from the convex process119883(119905) at times 1199051 and 1199052 Inview of the existence of the correlation a rotary ellipse modelis constructed to restrict the value range of 119883(1199051) and 119883(1199052)as shown in Figure 1(a) By regularization treatment (from119909 space to 119906 space) the standard model is then formed asillustrated in Figure 1(b)

Definition 2 If 119883(119905) is a convex process for any times 1199051 and1199052 the autocovariance function of interval variables119883(1199051) and119883(1199052) is defined as

Cov119883(1199051 1199052) = Cov (1198801 1198802) sdot 119883

119903

(1199051) sdot 119883119903

(1199052)

= (1199032119898minus 1) sdot 119883119903

(1199051) sdot 119883119903

(1199052)

0 le 119903119898le radic2

(3)

where 119903119898either stands for the major axis 1199031 of the ellipse in

119906 space if the slope of the major axis is positive or representsthe minor axis 1199032 if the slope of the major axis is negative Fordetails see Figure 2

Definition 3 With regard to the convex process 119883(119905) theautocorrelation coefficient function of 119883(1199051) and 119883(1199052) isexpressed as

120588119883(1199051 1199052) =

Cov119883(1199051 1199052)

radic119863119883(1199051) sdot radic119863119883

(1199052)=

Cov (1198801 1198802)

radic1198631198801sdot radic119863

1198802

= 12058811988011198802

= 1199032119898minus 1

(4)

where 1198631198801

and 1198631198802

denote the variance functions of thestandard interval variables 1198801 and 1198802 (119863

1198801= 119863

1198802=

1) 120588119883(1199051 1199052) is a dimensionless quantity and its magnitude

represents the linear correlation of 119883(1199051) and 119883(1199052) It isobvious that |120588

119883(1199051 1199052)| le 1 and 120588

119883(119905 119905) = 1

Several specific ellipse models with different values of120588119883(1199051 1199052) are clearly illustrated in Figure 3 Among all the

ellipses the one with 120588119883(1199051 1199052) = 0 has amaximal area which

implies the largest scattering degree and hence a minimumcorrelativity of variables 119883(1199051) and 119883(1199052) Moreover a larger|120588

119883(1199051 1199052)|means a stronger linear correlation between119883(1199051)

and 119883(1199052) Once |120588119883(1199051 1199052)| = 1 the ellipse model will bereplaced by a straight line that indicates a complete linearity

As mentioned above the characteristics of one time-varying uncertain parameter can be commendably embod-ied However with respect to complicated structures morecommon case is that we must solve the problem of multi-dimensional time-varying uncertainty The cross-correlationbetween two convex processes should be considered as well

Definition 4 Suppose two convex processes 119883(119905) and 119884(119905)for any times 1199051 and 1199052 the cross-correlation coefficientfunction of119883(1199051) and 119884(1199052) is established as

120588119883119884(1199051 1199052) =

Cov119883119884(1199051 1199052)

radic119863119883(1199051) sdot radic119863119884

(1199052)= 119903

2119898lowast minus 1

0 le 119903119898lowast le radic2

(5)

where Cov119883119884(1199051 1199052)means the cross-covariance function and

119903119898lowast represents the majorminor axis of the ellipse in 119906 space

derived from 119883(1199051) and 119884(1199052) (similarly to the definition in(3))

According to the above definitions the characteristicparameters of nonprobabilistic convex process model areavailable In the next section associated with the set-theorymethod the convex process model utilized to describe thetime-varying uncertainty of structural limit state is estab-lished


0

2

2 0

2

2 U1

U2

5∘120579 = minus4U1

U2

120579 = 45∘

r1

r1

r2

r2

rm = r1

rm = r2

(a) (b)

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

120588X(t1 t2) = 05

120588X(t1 t2) = 03

120588X(t1 t2) = 1

120588X(t1 t2) = 0

120588X(t1 t2) = 08

U1

U2

0

120588X(t1 t2) = minus05

120588X(t1 t2) = minus03

120588X(t1 t2) = minus1

120588X(t1 t2) = 0

120588X(t1 t2) = minus08

U1

U2

0

Figure 3 Typical geometric shapes of ellipses for different correlation coefficients

3 Time-Varying Uncertainty QuantificationAnalysis Corresponding to the Limit State

With regard to any engineering problems it is extremelysignificant to determine the performance of structural limitstate Once time-varying uncertainties originating frominherent properties or external environmental conditionsare considered the limit state should have time-variantuncertainty as well In fact if we define the time-varyingstructural parameters as convex processes the limit-statefunction can also be enclosed by a model of convex processThe detailed procedure of construction of the convex processmodel for structural limit state ismathematically discussed inthe following statements

A linear case of the limit-state function of the time-varying structure is expressed as

119892 (119905) = 119892 (119905 a (119905) X (119905)) = 1198860 (119905) +119899

sum

119894=1(119886

119894(119905) sdot 119883

119894(119905)) (6)

where X(119905) = (1198831(119905) 1198832(119905) 119883119899(119905))

119879 denotes a basicvector of convex processes and a(119905) = (1198860(119905) 1198861(119905) 1198862(119905) 119886

119899(119905))means a time-variant coefficient vector

By virtue of the set-theory method the mean valuefunction and the radius function of limit-state function 119892(119905)are given by


119892119888

(119905) = 119892119888

(119905 a (119905) X (119905)) = 1198860 (119905) +119899

sum

119894=1(119886

119894(119905) sdot 119883

119888

119894(119905))

119892119903

(119905) = radic119863119892(119905) = 119892

119903

(119905 a (119905) X (119905)) =radic

119899

sum

119894=1(119886

119894(119905) sdot 119883

119903

119894(119905))

2+

119899

sum

119894=1

119899

sum

119895=1119895 =119894

120588119883119894119883119895

(119905 119905) sdot 119886119894(119905) sdot 119886

119895(119905) sdot 119883

119903

119894(119905) sdot 119883

119903

119895(119905)

(7)

where Xc(119905) = (119883

119888

1(119905) 119883119888

2(119905) 119883119888

119899(119905))

119879 and Xr(119905) = (119883

119903

1(119905)

119883119903

2(119905) 119883119903

119899(119905))

119879 are respectively the vectors of mean valuefunction and the radius function for X(119905) 120588

119883119894119883119895

(119905 119905) standsfor the cross-covariance function of variables119883

119894(119905) and119883

119895(119905)

and119863119892(119905) is the variance function

In addition taking into account the comprehensive effectof the autocorrelation between119883

119894(1199051) and119883119894

(1199052) as well as thecross-correlation between 119883

119894(1199051) and 119883119895

(1199052) the autocovari-ance function corresponding to 119892(1199051) and 119892(1199052) is found to be

Cov119892(1199051 1199052) =

119899

sum

119894=1120588119883119894

(1199051 1199052) sdot 119886119894 (1199051) sdot 119886119894 (1199052) sdot 119883119903

119894(1199051)

sdot 119883119903

119894(1199052) +

119899

sum

119894=1

119899

sum

119895=1119895 =119894

120588119883119894119883119895

(1199051 1199052) sdot 119886119894 (1199051) sdot 119886119895 (1199052)

sdot 119883119903

119894(1199051) sdot 119883

119903

119895(1199052)

(8)

where 120588119883(1199051 1199052) is the autocorrelation coefficient function

of 119883119894(1199051) and 119883119894

(1199052) and 120588119883119894119883119895

(1199051 1199052) is the cross-covariancefunction of119883

119894(1199051) and119883119895

(1199052)In terms of the definitions in (5) we arrive at

120588119892(1199051 1199052) =

Cov119892(1199051 1199052)

radic119863119892(1199051) sdot radic119863119892

(1199052) (9)

It is instructive to discuss the case of 119892(119877(119905) 119878(119905)) = 119877(119905)minus119878(119905) in which the strength 119877(119905) and the stress 119878(119905) are bothenclosed by convex processes we get

120588119892(1199051 1199052) = [119877

119903

(1199051) sdot 119877119903

(1199052) sdot 120588119877 (1199051 1199052) + 119878119903

(1199051) sdot 119878119903

(1199052)

sdot 120588119878(1199051 1199052) minus 119877

119903

(1199051) sdot 119878119903

(1199052) sdot 120588119877119878 (1199051 1199052) minus 119877119903

(1199052) sdot 119878119903

(1199051)

sdot 120588119877119878(1199052 1199051)]

sdot1

radic[(119877119903 (1199051))2+ (119878119903 (1199051))

2minus 2119877119903 (1199051) sdot 119878

119903 (1199051) sdot 120588119877119878 (1199051 1199051)]

sdot1

radic[(119877119903 (1199052))2+ (119878119903 (1199052))

2minus 2119877119903 (1199052) sdot 119878

119903 (1199052) sdot 120588119877119878 (1199052 1199052)]

(10)

In accordance with the above equations the time-varyinguncertainty of limit-state 119892(119905) is explicitly embodied by anonprobabilistic convex process modelThat is to say for anytimes 1199051 and 1199052 once the time-variant uncertainties existingin structural parameters are known the ellipse utilized to

restrict the feasible domain between 119892(1199051) and 119892(1199052) can beaffirmed exclusively

Apparently it should be indicated that if the limit-state function is nonlinear several linearization techniquessuch as Taylorrsquos series expansion and interval perturbationapproach may make effective contributions to help us con-struct its convex process model

4 Time-Dependent Reliability Measure Indexand Its Solving Process

41 The Classical First-Passage Method in Random Pro-cess Theory For engineering problems we are sometimesmore interested in the calculation of time-varying probabil-itypossibility of reliability because it provides us with thelikelihood of a product performing its intended function overits service time From the perspective of randomness thetime-dependent probability of failuresafety during a timeinterval [0 119879] is computed by

119875119891(119879) = Pr exist119905 isin [0 119879] 119892 (119905X (119905)) le 0 (11)

or

119877119904(119879) = 1minus119875

119891(119879)

= Pr forall119905 isin [0 119879] 119892 (119905X (119905)) gt 0 (12)

where X(119905) is a vector of random processes and Pr119890 standsfor the probability of event 119890

Generally it is extremely difficult to obtain the exactsolution of 119875

119891(119879) or 119877

119904(119879) At present the most common

approach to approximately solve time-dependent reliabilityproblems in a rigorous way is the so-called first-passageapproach (represented schematically in Figure 4)

The basic idea in first-passage approach is that crossingfrom the nonfailure into the failure domain at each instanttimemay be considered as being independent of one anotherDenote by119873+

(0 119879) the number of upcrossings of zero-valueby the compound process 119892(119905X(119905)) from safe domain to thefailure domain within [0 119879] and hence the probability offailure also reads

119875119891(119879) = Pr (119892 (0X (0)) lt 0) cup (119873+

(0 119879) gt 0) (13)

Then the following upper bound on 119875119891(119879) is available

that is

119875119891(119879) le 119875

119891(0) +int

119879

0] (119905) 119889119905 (14)


T

0

g(t)

First-passage outcrossing

g(t) = g(X(t))

Figure 4 Schematic diagram for the first-passage approach

where 119875119891(0) is the probability of failure when time is fixed as

zero and ](119905) is the instantaneous outcrossing rate at time 119905The specific expression of ](119905) is evaluated as

] (119905)

= limΔ119905rarr 0

Pr (119892 (119905X (119905)) gt 0) cap (119892 (119905 + Δ119905X (119905 + Δ119905)) lt 0)Δ119905

(15)

Introducing the finite-difference concept rewrite (15) inthe form

] (119905)

asympPr (119892 (119905X (119905)) gt 0) cap (119892 (119905 + Δ119905X (119905 + Δ119905)) lt 0)

Δ119905

(16)

where Δ119905 has to be selected properly (under the sufficientlysmall level)

42 Time-Dependent Reliability Measurement Based on Non-probabilistic Convex Process Model Although the first-passage approach tends to provide a rigorous mathematicalderivation of the time-dependent reliability it is actually dif-ficult to evaluate the outcrossing rate for the general randomprocesses Only in few cases an analytical outcrossing rate isavailable Furthermore under the case of limited sample dataof time-varying uncertainty the analysis based on randomprocess theory is inapplicable Therefore in this section weapply the idea of the first-passage method into nonproba-bilistic convex process model and further establish a newmeasure index for evaluating time-dependent reliability withinsufficient information of uncertainty The details are asfollows

(i) Acquire convex process corresponding to the limit-state 119892(119905) (by reference to the corresponding Sections2 and 3)

(ii) Discretize 119892(119905) into a time sequence that is

119892 (0) 119892 (Δ119905) 119892 (2Δ119905) 119892 (119873Δ119905)

119873 =119879

Δ119905is sufficiently large

(17)

(iii) Construct each ellipse model with respect to 119892(119894Δ119905)and 119892((119894 + 1)Δ119905) and further investigate the interfer-ence conditions between elliptic domain and event119864

119894

119892(119894Δ119905) gt 0 cap 119892((119894 + 1)Δ119905) lt 0

(iv) Redefine the outcrossing rate ](119894Δ119905) as

] (119894Δ119905) asympPos 119864

119894

Δ119905

=Pos (119892 (119894Δ119905) gt 0) cap (119892 ((119894 + 1) Δ119905) lt 0)

Δ119905

(18)

where Pos119864119894 stands for the possibility of event 119864

119894

which is regarded as the ratio of the interference areato the whole elliptic area (119894 = 1 2 119873) that is

Pos 119864119894 = Pos (119892 (119894Δ119905) gt 0) cap (119892 ((119894 + 1) Δ119905) lt 0)

=119860119894

interference119860119894

total

(19)

(v) Determine the time-dependent reliability measureindex namely

119877119904(119879) = 1minus119875

119891(119879)

ge 1minus(Pos (0) +119873

sum

119894=1(] (119894Δ119905) sdot Δ119905))

= 1minus(Pos (0) +119873

sum

119894=1(Pos 119864

119894))

= 1minus(Pos (0) +119873

sum

119894=1(119860

iinterference119860119894

total))

(20)

43 Solution Strategy of Time-Dependent Reliability MeasureIndex Obviously the key point for determining theproposed time-dependent reliability measure index isjust the calculation of Pos119864

119894 (for practical problems of

(1199032radic2)119892119903

(119905) lt 119892119888

(119905) lt 119892119903

(119905) as shown in Figure 5(a))As mentioned above the regularization methodology isfirstly applied (as shown in Figure 5(b)) and the event 119864

119894is

equivalent to

119864119894 119892 (119894Δ119905) gt 0cap119892 ((119894 + 1) Δ119905) lt 0 997904rArr 119892

119888

(119894Δ119905)

+ 119892119903

(119894Δ119905) sdot 1198801 gt 0cap119892119888

((119894 + 1) Δ119905)

+ 119892119903

((119894 + 1) Δ119905) sdot 1198802 lt 0 997904rArr 1198801 gtminus119892119888

(119894Δ119905)

119892119903 (119894Δ119905)

cap1198802 ltminus119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(21)

For the sake of convenience one more coordinate trans-formation is then carried out and a unit circular domain


g((i + 1)Δt)

g((i + 1)Δt)

gc((i + 1)Δt)

g((i + 1)Δt)g(iΔt)0

gc(iΔt)

gr(iΔt)

g(iΔt) g(iΔt)

gr((i + 1)Δt)

U2 = minusgc((i + 1)Δt)

gr((i + 1)Δt)

U2

r1

0

r2

U1

U1 = minusgc(iΔt)

gr(iΔt)

(a) (b)

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

is eventually obtained (as shown in Figure 6) Equation (21)leads to

119864119894997904rArr 1198801 gtminus

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap1198802 ltminus

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

997904rArr 11990311205771 cos 120579 minus 11990321205772 sin 120579 gtminus119892119888

(119894Δ119905)

119892119903 (119894Δ119905)

cap 11990311205771 sin 120579 + 11990321205772 cos 120579 ltminus119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

997904rArr 1205772 lt119903111990321205771 +

radic21199032

sdot119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap 1205772 ltminus

119903111990321205771

minusradic21199032

sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(22)

From Figure 6 it becomes apparent that the shaded area119878119861119863119864

indicates the interference area 119860119894

interference and the totalarea119860119894

total always equals120587 Hence the analytical expression of119878119861119863119864

is of paramount importance and several following stepsshould be executed

Step 1 (solution of the coordinates of points 119860 and 119861) Let usconsider the simultaneous equations as

1198971 11990321205772 minus 11990311205771 minusradic2119892119888

(119894Δ119905)

119892119903 (119894Δ119905)= 0

1205771 = minusradic1 minus 12057722

(23)

By elimination method a quadratic equation is furtherdeduced that is

(11990321 + 119903

22) sdot 120577

22 minus 2radic21199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)sdot 1205772 + 2(

119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2

minus 11990321 = 0

(24)

1ζ

2ζ

O

A

D

C

B

E

R = 1d

Figure 6 Schematic diagram for the solution of time-dependentreliability measure index

Thus the coordinates of points 119860 and 119861 on 1205772-axis arerespectively

1205772 119860=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)

+radic(11990322 minus 2) sdot (

119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(25)

1205772 119861=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)

minusradic(11990322 minus 2) sdot (

119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(26)


Substituting (25) and (26) into (23) the coordinates ofpoints119860 and 119861 on 1205771-axis namely 1205771 119860

and 1205771 119861 can be easily

obtained

Step 2 (solution of the coordinates of points119862 and119863) Takinginto account the simultaneous equations

1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0


(27)

we arrive at

(11990321 + 119903

22) sdot 120577

22 + 2radic21199032 sdot

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

sdot 1205772

+ 2(119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2minus 119903

21 = 0

(28)

Therefore the coordinates of points 119862 and 119863 on 1205772-axis areevaluated as

1205772 119862=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(29)

1205772 119863=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(30)

Substituting (29) and (30) into (27) the coordinates ofpoints119862 and119863 on 1205771-axis namely 1205771 119862


determined

Step 3 (solution of the coordinates of point 119864) Since 119864

is actually the point of intersection between 1198971 and 1198972 thecoordinate calculation is satisfied


(119894Δ119905)

119892119903 (119894Δ119905)= 0

1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0(31)

Then we have1205771 119864

=minusradic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) + 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))

21199031

1205772 119864

=

radic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) minus 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))21199032

(32)

By virtue of the above three steps the geometric informa-tion of the characteristic points (from119860 to119864) can be obtainedmathematically and the coordinates of these points are thenused to deduce the analytical expression of119860119894

interference (119878119861119863119864)In fact the physical mean for definitions of the characteristicpoints lies in that the possibility of the cross failure during asmall time interval can be eventually embodied by a form ofgeometric domain which is determined by the points

Step 4 (solution of the fan-shaped area 11987810067040 119860119862119861119863

) The centralangel ang119860119874119863 is firstly computed by

ang119860119874119863 = tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

) (33)

The area 11987810067040 119860119862119861119863

equals

11987810067040 119860119862119861119863

=12ang119860119874119863 sdot 119877

2=12ang119860119874119863 sdot 12

=12(tanminus1 (

10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

))

(34)

where 119877 represents the radius of the unit circular domain

Step 5 (solution of the triangular areas 119878Δ119860119874119864

and 119878Δ119863119874119864

) Aspreviously noted the coordinates of points 119860 119863 and 119864 havebeen availableThus the triangular area 119878

Δ119860119874119864is derived from

119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119860

1205771 119864

0 1205772 1198601205772 119864

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119860

sdot 1205772 119864minus 1205771 119864

sdot 1205772 119860)

(35)

Similarly 119878Δ119863119874119864

is

119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119864

1205771 119863

0 1205772 1198641205772 119863

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119864

sdot 1205772 119863minus 1205771 119863

sdot 1205772 119864)

(36)

Step 6 (solution of the bow-shaped area 119878∙119860119862119861

) It can be foundthat the bow-shaped area 119878∙

119860119862119861is denoted by the difference

with the fan-shaped area 11987810067040 119860119862119861

and the triangular area 119878Δ119860119874119861

namely

119878∙

119860119862119861= 119878

10067040 119860119862119861minus 119878

Δ119860119874119861

11987810067040 119860119862119861

=12ang119860119874119861 sdot 119877

2=12ang119860119874119861 sdot 12

=12sdot 2 cosminus1119889 sdot 12 = cosminus1119889

119878Δ119860119874119861

=12119860119861 sdot 119889 =

12sdot 2radic1 minus 1198892 sdot 119889 = 119889radic1 minus 1198892

(37)

where 119860119861 is the chord length and 119889 as the distance fromorigin to line 1198971 equals 119892

119888

(119894Δ119905)119892119903

(119894Δ119905)


Time discretizationstrategy g(iΔt)

Solution of

+

Begin

End

NoYes

Ellipse model forFigure 5for details

Geometric domain forcross failure

Figure 6 for details

7 steps forEqs (23)~(38)

Eq (20)

i lt N

g(iΔt) and g((i + 1)Δt)

i = i + 1

solving PosEi

Posf(0)

Pf(T) and Rs(T)

N

sumi=1

(PosEi)

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

Pf(T) =NfailedNtotal

Rs(T) =Ntotal minus Nfailed

Ntotal

(i)

(ii)(iii)

(iv)

(v)

Initialization of Ntotal Δt T n= 0 Nfailed = 0 i = 1 j = 0

Discretization treatment

No

No

No

Yes

Yes

Yes

Yes

No

Yes YesXi Xi(0) Xi(Δt) Xi(NΔt)

N = TΔt

No

No

i = i + 1

j = 0

j = 0

i = i + 1

i = 1

i gt n

i gt nEstablishment of Ωi defined by (39)

Sample generationx i xi(0) xi(Δt)

xi(NΔt)

xi isin Ωi

Calculation of g(jΔt) byx(jΔt) x1(jΔt) x2(jΔt) xn(jΔt)

g(jΔt) gt 0

j lt N

j = j + 1

Nfailed = Nfailed + 1

Ngenerated le Ntotal

Ngenerated

Ngenerated = Ngenerated + 1

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

Step 7 (solution of Pos119864119894) Consider

Pos 119864119894 =

119860119894


total=119878119861119863119864

120587

=119878

10067040 119860119862119861119863minus 119878

Δ119860119874119864minus 119878

Δ119863119874119864minus 119878

∙

119860119862119861

120587

(38)

We should calculate Pos119864119894 successively (119894 = 1 2 119873)

based on the above steps By combination with the analysisstated in Section 42 an integral procedure for constructionof the time-dependent reliability measure index can beeventually achieved For ease of understanding one flowchartis further added (see Figure 7 for details)

5 Monte-Carlo Simulation Method

For comparisonrsquos purpose this paper also provides a Monte-Carlo simulation method to compute structural time-dependent reliability The detailed analysis procedure isexpounded below (as is seen in Figure 8)

(i) Initialize the numbers of generated samples and failedsamples as 119873generated = 0 119873failed = 0 define the totalnumber of samples as119873total the cyclic counting index119895 is set to zero

(ii) Assume a small time increment Δ119905 and discretizeeach convex process 119883

119894(119905) as a time series X

119894

119883119894(0) 119883

119894(Δ119905) 119883

119894(119873Δ119905)119873 = 119879Δ119905 in sequence

(119894 = 1 2 119899) based on the characteristic propertiesof time-variant uncertainty included by 119883119888

119894(119905) 119883119903

119894(119905)

and 120588119883119894

(1199051 1199052) construct a (119873 + 1)-dimensionalellipsoidal domainΩi that is


Ωi

(Xi minusXci )

119879

sdot(

11986211 11986212 1198621119873+1

11986221 11986222 1198622119873+1

119862119873+11 119862

119873+12 119862119873+1119873+1

)

minus1

(Xi minusXci )

le 1

119894 = 1 2 119899

(39)

where 119862119897119904= Cov

119883119894

((119897 minus 1)Δ119905 (119904 minus 1)Δ119905)(iii) According to the ldquo3120590 rulerdquo generate one group of

normal distributed samples xi 119909119894(0) 119909

119894(Δ119905)

119909119894(119873Δ119905) subjected to each time series Xi (119894 =

1 2 119899) if xi could not meet the requirementsof inequation from (39) restart (iii) otherwise119873generated = 119873generated + 1 and then

(1) if119873generated le 119873total go to (iv)(2) if not go to (v)

(iv) Input the samples x(119895Δ119905) 1199091(119895Δ119905) 1199092(119895Δ119905) 119909119899(119895Δ119905) into the limit-state function 119892(119895Δ119905) to esti-

mate the structural safety at time 119895Δ119905 if 119892(119895Δ119905) gt 0and 119895 lt 119873 119895 = 119895 + 1 and restart (iv) else

(1) if 119895 lt 119873119873failed = 119873failed + 1 reset 119895 to zero andgo back to (iii)

(2) if not reset 119895 to zero and go back to (iii)

(v) The structural failure possibility is approximatelycomputed as 119875

119891(119879) = 119873failed119873total furthermore

the time-dependent reliability is obviously depicted as119877119904(119879) = (119873total minus 119873failed)119873total

By means of the proposed methodology of Monte-Carlosimulations the numerical results of time-dependent relia-bility can be obtained in principle However some importantissues remain unsolved in practical applications (1) thereis high complexity in sample generation for all convexprocesses which is mainly embodied by the construction ofmultidimensional ellipsoidal convexmodels (Ω1Ω2 Ω119899)under the case of small time increment Δ119905 (2) The aboveanalysis is not able to demonstrate the cross-correlationbetween two convex processes (3) The given assumption ofldquo3120590 rulerdquomay result in an extremely large errorwhen encoun-tering the case of the combination of various uncertaintycharacteristics (4) Enormous computational costs have to

d1

d2

F1(t)F2(t)

F3(t)l

Figure 9 A cantilever beam structure

be confronted when solving complex engineering problemsTherefore compared with the Monte-Carlo simulations thepresented method based on nonprobabilistic convex processmodel may show superiority to some extent when dealingwith the problems of time-dependent reliability evaluation

6 Numerical Examples

In this section three engineering examples are investigatedAmong them the first two examples of a cantilever beamstructure and a ten-bar truss structure are further analyzedbyMonte-Carlo simulations and hence the numerical resultscan be regarded as a reference to effectively demonstrate thevalidity and feasibility of the presented methodology That isto say the accuracy of the proposed method is verified bythe simulation techniques and the deviation of the reliabilityresults may quantify a specific precision level

Additionally the safety estimation of a complicated pro-peller structure in the last example can better illustrate theadvantage and capability of the developed time-dependentreliability method when tackling reliability issues of largecomplex structures

61 ACantilever BeamStructure Acantilever beam structuremodified from thenumerical example in [28] is considered asshown in Figure 9 Three time-varying external forces 1198651(119905)1198652(119905) and 1198653(119905) are applied to the beam and the maximummoment on the constraint surface at the origin should be lessthan an allowable value119872criteria(119905) Thus the following limit-state function can be created by

119892 (119905) = 119892 (1198651 (119905) 1198652 (119905) 1198653 (119905) 119872criteria (119905))

= 119872criteria (119905) minus 1198891 sdot 1198651 (119905) minus 1198892 sdot 1198652 (119905) minus 119897

sdot 1198653 (119905) 119905 isin [0 119879]

(40)

where 1198891 = 1m 1198892 = 2m and 119897 = 5m and 119879 in this problemis defined as 10 years

Here 1198651(119905) 1198652(119905) 1198653(119905) and119872criteria(119905) are treated as con-vex processes containing autocorrelation but without consid-eration of cross-correlation All the uncertainty propertiesare summarized in Table 1 where 120572 = 1 12 14 16 18 2and 119896 = 1 2 3 4 5 Various combinations of 120572 and 119896

embody different spans and autocorrelation effects of convexprocesses

The time-dependent reliability results 119877119904(119879) obtained

by the nonprobabilistic convex process model are given inTable 2 and Figure 10 (Δ119905 = 005 years) It can be foundthat values of 119877

119904(119879) decrease remarkably when either 120572 or 119896


Table 1 Time-varying uncertainty characteristics of the cantilever beam structure

Mean value functions and radius functions of convex processes 1198651(119905) 119865

2(119905) 119865

3(119905)119872criteria(119905)

119865119888

1(119905) =

1

2sin(4120587119905 + 120587

4) +

5

2

119865119903

1(119905) = 119865

119888

1(119905) times 6120572

(unit N)

119865119888

2(119905) = minus cos(8120587119905 minus 120587

3) + 5

119865119903

2(119905) = 119865

119888

2(119905) times 5120572

(unit N)

119865119888

3(119905) = minus

3

4cos(6120587119905 + 2120587

5) +

3

2

119865119903

3(119905) = 119865

119888

3(119905) times 3120572

(unit N)

119872119888

criteria (119905) = 28 minus1

5119905

119872119903

criteria (119905) =120572119905

2

100

(unit Nsdotm)Correlation coefficient functions of convex processes 119865

1(119905) 119865

2(119905) 119865

3(119905)119872criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(4119896sdot|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890

minus(5119896sdot|t1minust2 |) sdot cos (119896 sdot 1003816100381610038161003816t1 minus t21003816100381610038161003816) No consideration of

cross-correlation1205881198653

(1199051 119905

2) = 119890

minus((52)119896sdot|t1minust2 |)2 120588119872criteria

(1199051 119905

2) =

1

2(1 + cos (6119896 sdot 1003816100381610038161003816t1 minus t2

1003816100381610038161003816))

Table 2 Reliability results based on the nonprobabilistic convexprocess model

119896120572 120572 = 10 120572 = 12 120572 = 14 120572 = 16 120572 = 18 120572 = 20119896 = 1 1 0986 09426 08931 08259 07595119896 = 2 1 09857 09424 08881 08155 07425119896 = 3 1 09854 09423 08821 08027 07249119896 = 4 1 09853 09419 08741 07915 07085119896 = 5 1 09852 09406 08696 07844 06972

10 2018161412070

075

080

085

090

095

100

Based on the nonprobabilistic convex process model

Tim

e-de

pend

ent r

eliab

ility

k = 1 k = 2k = 3

k = 4k = 5

120572 (T = 10 years)

Figure 10 Time-dependent reliability results versus 120572 and 119896

increases as expected This indicates that higher dispersionor weaker correlation leads to lower structural safety Forexample the cantilever beam is absolutely safe when 120572 and119896 are both equal to 1 while 119877

119904(119879) only reaches 06972 under

the case of 120572 = 2 and 119896 = 5To better analyze the accuracy of the time-dependent

reliability results the Monte-Carlo simulation method is alsoused to deal with a severe case of 120572 = 2 (100000 samples)Thecomparative results are illustrated in Figure 11 As shown inFigure 11 119877

119904(119879) calculated by our method is coincident with

the results derived from the numerical simulations However

1 2 3 4 5068

070

072

074

076

078

080

Tim

e-de

pend

ent r

eliab

ility

Obtained by the nonprobabilistic convex process modelObtained by the Monte-Carlo simulations

Different values of k (120572 = 2)

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

there are three points that should be noted (1) The resultsbased on convex process model are more conservative dueto the fact that less information on time-varying uncertaintyis needed (2) Associated with the increasing values of 119896the deviation of the reliability results 119877

119904(119879) is widening

(from 290 to 592) (3) The accuracy of the reliabilityfrom Monte-Carlo simulations mainly relies on the numberof samples which implies that the situation of intensivecomputation and consuming timemay have to be confronted

62 A Ten-Bar Truss Structure A well-known ten-bar trussstructure modified from the numerical example in [15] isinvestigated as depicted in Figure 12 Youngrsquos modulus 119864 =

68948MPa and the density 120588 = 785 times 10minus9 119905mm3 Thelength 119897 of the horizontal and vertical bars is 9144mm andthe area of each bar equals 4000mm2 Subjected to two time-varying vertical forces 1198651(119905) and 1198652(119905) and one time-varyinghorizontal force 1198653(119905) the structure is defined as failure if119878119879max(119905) ge 119877

119879criteria(119905) or |119878119862max(119905)| ge |119877119862criteria(119905)| occurs

where 119878119879max(119905) and 119878119862max(119905) stand for the maximal values of

tensile stress and compressive stress among all the members


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


2(119905) 119865

3(119905) 119877

119879criteria(119905) 119877119862criteria(119905)

119865119888

1(119905) = 119865

119888

2(119905) = 2 sin(3120587119905 + 120587

3) + 4448

119865119903

1(119905) = 119865

119903

2(119905) = 119865

119888

1(119905) times 3120572

(unit KN)

119865119888

3(119905) = 7 cos (4120587119905 minus 120587

5) + 17792

119865119903

3(119905) = 119865

119888

3(119905) times 2120572

(unit KN)

119877119888

119879criteria(119905) = 375 minus15119905

8119877119903

119879criteria(119905) =120572119905

2

36

119877119888

119862criteria(119905) = minus100 +3119905

4119877119903

119862criteria(119905) =120572119905

2

32

(unit MPa)Correlation coefficient functions of convex processes 119865

1(119905) 119865

2(119905) 119865

3(119905) 119877


1205881198651

(1199051 119905

2) = 119890

minus(5|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890

minus(5|t1minust2 |) sdot cos (4 1003816100381610038161003816t1 minus t21003816100381610038161003816) Taking into account the cross-correlation between

1198651(119905) and 119865

2(119905)120588

1198653

(1199051 119905

2) = 119890

minus(6|t1minust2 |)2 120588119877119879criteria

(1199051 119905

2) = 120588

119877119862criteria

(1199051 119905

2) = 0

1

246

35

l l

X

Y

43

108

65

97

21

l

F1(t) F2(t)

F3(t)

Figure 12 A ten-bar truss structure

119877119879criteria(119905) and 119877

119862criteria(119905) are respectively the allowablevalues at time 119905 under the state of tension and compressionHence the following two limit-state functions are expressedas1198921 (119905) = 119892 (119878119879max (119905) 119877119879criteria (119905))

= 119877119879criteria (119905) minus 119878119879max (1198651 (119905) 1198652 (119905) 1198653 (119905))

119905 isin [0 119879]

(41)

1198922 (119905) = 119892 (119878119862max (119905) 119877119862criteria (119905))

=1003816100381610038161003816119877119862criteria (119905)

1003816100381610038161003816

minus1003816100381610038161003816119878119862max (1198651 (119905) 1198652 (119905) 1198653 (119905))

1003816100381610038161003816

119905 isin [0 119879]

(42)

where 119879 = 10 yearsIn this problem 1198651(119905) 1198652(119905) 1198653(119905) 119877119879criteria(119905) and

119877119862criteria(119905) are defined as convex processes where the auto-

correlation and the cross-correlation between 1198651(119905) and 1198652(119905)are simultaneously considered The uncertainty properties ofthe above convex processes are given in Table 3 where 120572 =

15 16 17 18 19 2 and the cross-correlation coefficient12058811986511198652

= minus02 01 0 01 02It is not difficult to understand that member 4 and

member 8 are most dangerous respectively under the ten-sile situation and compressive situation According to thedefinitions in (41) and (42) the time-dependent reliabilityresults119877

119904(119879) obtained by the nonprobabilistic convex process

Table 4 Reliability results based on the nonprobabilistic convexprocess model (member 4)

12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 09096 0873 08329 07904 07369 0671412058811986511198652

= minus01 09085 08704 08319 07873 07327 0664712058811986511198652

= 0 0906 08684 08302 07842 07284 0663512058811986511198652

= 01 09053 08675 08296 07821 07272 065912058811986511198652

= 02 09047 08669 08276 07805 07238 06579


12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 1 09947 09581 08928 07899 0652112058811986511198652

= minus01 099941 09817 0929 08385 07075 0565812058811986511198652

= 0 09954 09603 08888 07762 0633 048212058811986511198652

= 01 09859 0935 08405 07057 05589 0402812058811986511198652

= 02 09706 09003 07858 06381 04884 03216

model are given in Tables 4 and 5 and Figures 13 and 14(Δ119905 = 005 years) It is remarkable that (1) the results of119877119904(119879) continue to decline as the increase of 119886 (2) If 120588

11986511198652is

negative stronger cross-correlationmay lead to a higher levelof structural safety if 120588

11986511198652is positive however the truss will

be more dangerous with an increasing 12058811986511198652

(3) 12058811986511198652

has aminor effect on reliability results for member 4 while it playsan important role in safety estimation for member 8

For the purpose of verification and comparison theMonte-Carlo simulation method is utilized again to analyzethe case of 120588

11986511198652= 0 (in disregard of cross-correlation) The

time-dependent reliability results are illustrated in Figures 15and 16 Similarly to conditions on the first example the time-dependent reliability results calculated by convex processmodel are consistent with the results computed by theMonte-Carlo simulation method in qualitative analysis but lowerto some extent from quantitative perspective Specificallywhen 119886 = 2 the maximal deviations of reliability results arerespectively 1306 for member 4 and 2617 for member 8

63 A Marine Propeller Structure The high-speed rotatingpropeller structure as a vital component of power plantmust reach high standard of safety during its whole lifetime


15 2019181716065

070

075

080

085

090


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)

to guarantee normal operations of marine system In actualworking conditions affected by changeable environment ofwater flow the forces experienced on propeller are alwayscomplicated and are uncertain at different times Thereforethe time-dependent reliability analysis of this structure is ofgreat significance Figure 17 shows a specific type of propellerwhich is composed of four blades and one support block andits material parameters are listed in Table 6 Three types oftime-varying forces are applied to the surface of each blade

15 2019181716060

065

070

075

080

085

090

095

Tim

e-de

pend

ent r

eliab

ility


(120588F1F2 = 0)Different values of 120572

Figure 15 Comparisons of the reliability results obtained by theconvex process model and the Monte-Carlo based model (member4)

15 2019181716

04

05

06

07

08

09

10

Tim

e-de

pend

ent r

eliab

ility




namely the propulsive force 1198651(119905) the centrifugal force 1198652(119905)and the shear force 1198653(119905) Hence the limit-state function isexpressed as

119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported

Propulsive force F1(t)

Centrifugal force F2(t)

Shear force F3(t)

Figure 17 A marine propeller structure

Table 6 Material parameters of the marine propeller structure

Youngrsquos modulus 119864 Poissonrsquos ratio ] Density 1205882 times 10

5 MPa 03 78 times 103 Kgm3

where 120590max(119905) denotes the maximum von Mises stress ofthe propeller structure 120590criteria(119905) is allowable strength and119879 = 10 years Considering that the propeller belongs to acomplicated three-dimensional structure the dynamic finiteelement model containing 44888 elements and 31085 nodesis set up to estimate the structural safety

Assume that dynamic forces 1198651(119905) 1198652(119905) and 1198653(119905) aswell as allowable strength 120590criteria(119905) are all convex processeswhere autocorrelation and cross-correlation are both con-sidered The detailed information of time-varying uncer-tainty is summarized in Table 7 where 120572 can be valued by01 02 1

The present method is applied to the above engineeringproblem and all the analysis results are given in Table 8 andFigure 18 (Δ119905 = 005 years) It can be seen that (1) there is noneed to worry about failure when 120572 = 01 or 02 (119877

119904(119879) = 1)

(2) As 120572 increases the time-dependent reliability may showa decreasing trend (the minimum 119877

119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

Figure 18 Time-dependent reliability results versus 120572

case of 120572 = 1) and its variation is essentially linear at thebeginning and then becomes nonlinear to a certain degree


Table 7 Time-varying uncertainty characteristics of the marine propeller structure


2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903

criteria(119905) = (119905 + 1) sdot 119890(t10) (Unit MPa)

Correlation coefficient functions of convex processes 1198651(119905) 119865

2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890

minus(2|t1minust2 |) sdot cos (6 1003816100381610038161003816t1 minus t21003816100381610038161003816) 120588

1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890

minus|t1minust2 | sdot cos (120587119905) + cos (3120587119905)) 12058811986521198653

(1199051 119905

2) =

8 minus 119905

20

Table 8 Reliability results based on the nonprobabilistic convex process model

120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538

64 Discussions on the Computational Results Synthesizingthe computational results of the above three numericalexamples the following points can be inherited

(1) The characteristics of the time-varying uncertainparameters generally exert a great influence on struc-tural safety On the one hand the increasing disper-sion and decreasing autocorrelation may result in asevere situation of reduced time-dependent reliability(mainly consulted by Section 61) on the other handthe role of cross-correlation between time-varyinguncertain parameters makes analysis more compli-cated (referred by Section 62)

(2) Because less assumptions of uncertainty are neededmore conservative reliability results given by presentmethod appear than those from numerical simula-tions However it should be emphasized that thestructural reliability is closely related to the uncertainparameters and hence subjective assumptions mayyield unreliable results especially when dealing withthe engineering cases of limited samples

(3) Directed at simple problems of dynamics the Monte-Carlo simulation method as an optional way canevaluate structural reliability by means of sufficientsamples and cumulative operations (such as Sec-tions 61 and 62) but is powerless when tack-ling mechanical problems which contain multi-dimensional uncertainties cross-correlation large-scale configurations or complex boundary conditions(as stated in Section 63 in this study)

In summary the numerical examples demonstrate thatthe time-dependent reliability analysis based on nonproba-bilistic convex process model has high efficiency and simul-taneously an acceptable analysis precisionMore importantlywe can provide a feasible and reasonable way to mathemati-cally evaluate the dynamical safety for complex engineeringproblems

7 Conclusions

With the rapid technological advance the reliability analysisconsidering time-varying effect has attracted more and more

concerns and discussions Currently most of the approachesfor performing time-dependent reliability assessment arealways based upon the random process model where thedistributions of time-varying uncertain parameters shouldbe determined by a substantial number of samples whichhowever are not always available or sometimes very costlyfor practical problems Thus some assumptions on distri-bution characteristics have to be made in many dynami-cal cases when using the probability model Neverthelessunjustified assumptions may give rise to misleading resultsunexpectedly

In view of the abovementioned facts this paper describesthe time-varying uncertainty with the model of nonprob-abilistic convex process In this convex process model theuncertain variables at any time are expressed as intervalsand the corresponding autocovariance function and auto-correlation coefficient function are established to charac-terize the relationship between variables at different timesReferred by the definitions in random process theory thecross-correlation between different two convex processes isalso considered Then by using the set-theory method andthe regularization technique the time-varying limit-statefunction is transformed and quantified by a standard convexprocess model with autocorrelation Enlightened by the ideasof first-passage method and static reliability analysis a newnonprobabilistic measurement of time-dependent reliabilityis proposed and its analytical expression in linear case isconducted mathematically Additionally as a means of verifi-cation and comparison the Monte-Carlo simulation methodis also presented and applied into the solution of numer-ical examples Analytical results indicate that the presentmethod can be ensured to be more applicable and efficientwhen estimating structural safety of complex engineeringproblems

Indeed the present time-dependent reliability analysistechnique based on the nonprobabilistic convex processmodel can be regarded as a beneficial supplement to thecurrent reliability theory of random processes On the basisof the reliability analysis the proposed safety measurementindex can be also applied to the fields of time-varyingstructural design optimization


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the National Nature ScienceFoundation ofChina (no 11372025) for the financial supportsBesides the authors wish to express their many thanks to thereviewers for their useful and constructive comments

References

[1] C Andrieu-Renaud B Sudret and M Lemaire ldquoThe PHI2method a way to compute time-variant reliabilityrdquo ReliabilityEngineering and System Safety vol 84 no 1 pp 75ndash86 2004

[2] H O Madsen S Krenk and N C LindMethods of StructuralSafety Dover New York NY USA 2006

[3] M Lemaire Structural Reliability ISTE-Wiley New York NYUSA 2009

[4] Z Wang and P Wang ldquoA new approach for reliability analysiswith time-variant performance characteristicsrdquoReliability Engi-neering amp System Safety vol 115 pp 70ndash81 2013

[5] A M Haofer ldquoAn exact and invariant first-order reliabilityformatrdquo Journal of Engineering Mechanics vol 100 pp 111ndash1211974

[6] J F Zhang and X P Du ldquoA second-order reliability methodwith first-order efficiencyrdquo Journal of Mechanical Design Trans-actions of the ASME vol 132 no 10 Article ID 101006 2010

[7] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[8] Y Ben-Haim ldquoA non-probabilistic concept of reliabilityrdquo Struc-tural Safety vol 14 no 4 pp 227ndash245 1994

[9] Y J Luo Z Kang and A Li ldquoStructural reliability assessmentbased on probability and convex set mixed modelrdquo Computersamp Structures vol 87 no 21-22 pp 1408ndash1415 2009

[10] C Jiang G Y Lu X Han and L X Liu ldquoA new reliability anal-ysis method for uncertain structures with random and intervalvariablesrdquo International Journal of Mechanics and Materials inDesign vol 8 no 2 pp 169ndash182 2012

[11] G Barone and D M Frangopol ldquoReliability risk and lifetimedistributions as performance indicators for life-cycle mainte-nance of deteriorating structuresrdquo Reliability Engineering andSystem Safety vol 123 pp 21ndash37 2014

[12] J Li J-B Chen and W-L Fan ldquoThe equivalent extreme-value event and evaluation of the structural system reliabilityrdquoStructural Safety vol 29 no 2 pp 112ndash131 2007

[13] ZWang Z P Mourelatos J Li I Baseski and A Singh ldquoTime-dependent reliability of dynamic systems using subset simula-tion with splitting over a series of correlated time intervalsrdquoJournal of Mechanical Design vol 136 Article ID 061008 2014

[14] M Mejri M Cazuguel and J Y Cognard ldquoA time-variantreliability approach for ageing marine structures with non-linear behaviourrdquoComputers amp Structures vol 89 no 19-20 pp1743ndash1753 2011

[15] G Tont L Vladareanu M S Munteanu and D G TontldquoMarkov approach of adaptive task assignment for roboticsystem in non-stationary environmentsrdquoWseas Transactions onSystems vol 9 no 3 pp 273ndash282 2010

[16] V Bayer and C Bucher ldquoImportance sampling for first passageproblems of nonlinear structuresrdquo Probabilistic EngineeringMechanics vol 14 no 1-2 pp 27ndash32 1999

[17] S O Rice ldquoMathematical analysis of random noiserdquo The BellSystem Technical Journal vol 23 pp 282ndash332 1944

[18] E H Vanmarcke ldquoOn the distribution of the first-passage timefor normal stationary random processesrdquo Journal of AppliedMechanics Transactions ASME vol 42 no 1 pp 215ndash220 1975

[19] P H Madsen and S Krenk ldquoAn integral equation methodfor the first-passage problem in random vibrationrdquo Journal ofApplied Mechanics vol 51 no 3 pp 674ndash679 1984

[20] Z Hu and X Du ldquoTime-dependent reliability analysis withjoint upcrossing ratesrdquo Structural and Multidisciplinary Opti-mization vol 48 no 5 pp 893ndash907 2013

[21] G Schall M H Faber and R Rackwitz ldquoThe ergodicityassumption for sea states in the reliability estimation of offshorestructuresrdquo Journal of Offshore Mechanics and Arctic Engineer-ing vol 113 no 3 pp 241ndash246 1991

[22] S Engelund R Rackwitz and C Lange ldquoApproximations offirst-passage times for differentiable processes based on higher-order threshold crossingsrdquo Probabilistic Engineering Mechanicsvol 10 no 1 pp 53ndash60 1995

[23] H Streicher and R Rackwitz ldquoTime-variant reliability-orientedstructural optimization and a renewal model for life-cyclecostingrdquo Probabilistic Engineering Mechanics vol 19 no 1 pp171ndash183 2004

[24] Q Li C Wang and B R Ellingwood ldquoTime-dependentreliability of aging structures in the presence of non-stationaryloads and degradationrdquo Structural Safety vol 52 pp 132ndash1412015

[25] Z P Mourelatos M Majcher V Pandey and I BaseskildquoTime-dependent reliability analysis using the total probabilitytheoremrdquo Journal of Mechanical Design vol 137 no 3 ArticleID 031405 2015

[26] B Sudret ldquoAnalytical derivation of the outcrossing rate intime-variant reliability problemsrdquo Structure and InfrastructureEngineering vol 4 no 5 pp 353ndash362 2008

[27] C Jiang B Y Ni X Han and Y R Tao ldquoNon-probabilisticconvex model process a new method of time-variant uncer-tainty analysis and its application to structural dynamic relia-bility problemsrdquo Computer Methods in Applied Mechanics andEngineering vol 268 pp 656ndash676 2014

[28] X Wang L Wang I Elishakoff and Z Qiu ldquoProbability andconvexity concepts are not antagonisticrdquo Acta Mechanica vol219 no 1-2 pp 45ndash64 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


[18] Madsen and Krenk developed an integral equationmethod for solving the first-passage problems [19] a time-dependent reliability analysis method with joint upcrossingrates inspired by [19] was further developed by Hu and Du[20] for more general cases of the limit-state functions thatinvolve time random variables and stochastic processes bycombination of the ideas of outcrossing and system reliabilityWang et al [13] presented an improved subset simulationwith splitting approach by partitioning the original highdimensional random process into a series of correlatedshort duration low dimensional random processes severalimproved formulations for calculating the outcrossing ratebased on the Poisson assumptionwere respectively extendedby Schall et al [21] Engelund et al [22] Streicher andRackwitz [23] and so forth Recently additional correlationalresearches have been also suggested by [24 25]

Compared to static reliability analysis the research ontime-dependent reliability is still in its preliminary stage andthe following features contribute to the sources of difficulty(1) from the analytical point of view very few modelsor analysis methods have been presented in the past forthe evaluation of time-dependent reliability considering thegeneral case of non-Gaussianity nonstationarity and non-linear dependency (2) from the perspective of simulationextremely high computational cost is needed to guaranteereasonable numerical results of time-dependent reliabilitywhich greatly restrict the application in complex structuralsystems In recent developments the work of Sudret et al[1 26] where an analytical outcrossing rate utilizing the PHI2method is derived may partly overcome the abovementionedinsufficiencies

As the literature survey reveals most of the existingstudies focus on the random process models when perform-ing the time-dependent reliability analysis while ignoringthe existence of nonprobabilistic time-varying uncertaintiesFurthermore the assessment of the time-dependent relia-bility by random process theory must require knowledge ofprobabilistic descriptions for all time-varying uncertaintieswhich are typically determined by sufficient experimentalsamples Unfortunately for practical engineering problemsexperimental samples are not always available or re some-times very expensive to obtain so that one cannot directlyestablish the precise analytical model Giving subjectiveassumptions for description of the uncertainty characteristicsis likely to bring about a serious error of the time-dependentreliability results In view of the above reasons it is quitenecessary and urgent to carry out time-dependent reliabilityresearch on nonprobabilistic modeling methods Jiang et al[27] proposed a new theory for time-varying uncertaintynamely the ldquonon-probabilistic convex process modelrdquo totackle the problems of safety estimate when lacking relevantuncertainty information However the solution of the time-dependent reliability must rely on the Monte-Carlo simu-lations rather than constructing an analytical modelindexAdditionally the work in [27] mainly concentrated on thecase of stationary convex process the more common case ofnon-stationary process was not discussed

Generally speaking compared with static reliability anal-ysis fewer studies on time-dependent reliability analysis are

performed at present and its theoretical foundation is stillrelatively immature In this paper inspired by the work in[26 27] we develop a new time-dependent reliability anal-ysis approach based on the nonprobabilistic convex processmodel in which a new measure index of time-dependentreliability is established and its solution procedure is deducedmathematically The presented approach can effectively dealwith the more general case containing nonstationarity andnonlinear dependency that is of particular importance insolving practical engineering problems

The paper is organized as follows Section 2 providesa brief review of general concept of nonprobabilistic con-vex process model and its relevant mathematical basis InSection 3 combinedwith the foregoing convex processmodeland the set-theory method the time-varying uncertaintyof structural limit state is quantified Section 4 details thepresented time-dependent reliability method and then weintroduce the Monte-Carlo simulations in Section 5 Theproposed methodology is demonstrated with three casestudies in Section 6 and then followed by conclusions inSection 7

2 Notation and Classification ofNonprobabilistic Convex Process Model

Enlightened by [27] the nonprobabilistic convex processmodel is introduced in this section to quantify the time-varying uncertain parameters in dynamic mechanics whenfacing the case of limited sample information Here theuncertainty of structural parameters at any time is depictedwith a bounded closed interval and the correlation betweenvariables in different instant time points is reflected by defin-ing a corresponding correlation function If we discretize thecontiguous convex process into a time series the feasibledomain of all interval variables belongs to a convex set Fordetails see the following definitions

Definition 1 Consider a convex process 119883(119905) 119883(119905) and 119883(119905)describe the upper and lower bound functions of 119883(119905) andhence the mean value function119883119888

(119905) and the radius function119883

119903

(119905) are respectively given by

119883119888

(119905) =

119883 (119905) + 119883 (119905)

2

119883119903

(119905) =

119883 (119905) minus 119883 (119905)

2

(1)

For convenience we also define the variance function119863

119883(119905) as

119863119883(119905) = (119883

119903

(119905))2= (

119883 (119905) minus 119883 (119905)

2)

2

(2)

It is apparently indicated that the properties of uncertainvariables at any time can be quantified mathematically from


0

(a) (b)

0

02

2

X(t2)

X(t2)

Xc(t2)

X(t2)


Xr(t1)

Xr(t2)

U1

U2





Cov119883(1199051 1199052) = Cov (1198801 1198802) sdot 119883

119903

(1199051) sdot 119883119903

(1199052)

= (1199032119898minus 1) sdot 119883119903

(1199051) sdot 119883119903

(1199052)

0 le 119903119898le radic2

(3)




120588119883(1199051 1199052) =

Cov119883(1199051 1199052)

radic119863119883(1199051) sdot radic119863119883

(1199052)=

Cov (1198801 1198802)


1198802

= 12058811988011198802

= 1199032119898minus 1

(4)

where 1198631198801

and 1198631198802


1198801= 119863

1198802=



119883(1199051 1199052)| le 1 and 120588

119883(119905 119905) = 1








120588119883119884(1199051 1199052) =

Cov119883119884(1199051 1199052)

radic119863119883(1199051) sdot radic119863119884

(1199052)= 119903



(5)






0

2

2 0

2

2 U1

U2

5∘120579 = minus4U1

U2

120579 = 45∘

r1

r1

r2

r2

rm = r1

rm = r2

(a) (b)


120588X(t1 t2) = 05

120588X(t1 t2) = 03

120588X(t1 t2) = 1

120588X(t1 t2) = 0

120588X(t1 t2) = 08

U1

U2

0

120588X(t1 t2) = minus05

120588X(t1 t2) = minus03

120588X(t1 t2) = minus1

120588X(t1 t2) = 0

120588X(t1 t2) = minus08

U1

U2

0





119892 (119905) = 119892 (119905 a (119905) X (119905)) = 1198860 (119905) +119899

sum

119894=1(119886

119894(119905) sdot 119883

119894(119905)) (6)

where X(119905) = (1198831(119905) 1198832(119905) 119883119899(119905))





119892119888

(119905) = 119892119888

(119905 a (119905) X (119905)) = 1198860 (119905) +119899

sum

119894=1(119886

119894(119905) sdot 119883

119888

119894(119905))

119892119903

(119905) = radic119863119892(119905) = 119892

119903

(119905 a (119905) X (119905)) =radic

119899

sum

119894=1(119886

119894(119905) sdot 119883

119903

119894(119905))

2+

119899

sum

119894=1

119899

sum

119895=1119895 =119894

120588119883119894119883119895

(119905 119905) sdot 119886119894(119905) sdot 119886

119895(119905) sdot 119883

119903

119894(119905) sdot 119883

119903

119895(119905)

(7)

where Xc(119905) = (119883

119888

1(119905) 119883119888

2(119905) 119883119888

119899(119905))

119879 and Xr(119905) = (119883

119903

1(119905)

119883119903

2(119905) 119883119903

119899(119905))


119883119894119883119895


119894(119905) and119883

119895(119905)



119894(1199051) and119883119894


119894(1199051) and 119883119895


Cov119892(1199051 1199052) =

119899

sum

119894=1120588119883119894

(1199051 1199052) sdot 119886119894 (1199051) sdot 119886119894 (1199052) sdot 119883119903

119894(1199051)

sdot 119883119903

119894(1199052) +

119899

sum

119894=1

119899

sum

119895=1119895 =119894

120588119883119894119883119895

(1199051 1199052) sdot 119886119894 (1199051) sdot 119886119895 (1199052)

sdot 119883119903

119894(1199051) sdot 119883

119903

119895(1199052)

(8)


of 119883119894(1199051) and 119883119894

(1199052) and 120588119883119894119883119895


119894(1199051) and119883119895


120588119892(1199051 1199052) =

Cov119892(1199051 1199052)

radic119863119892(1199051) sdot radic119863119892

(1199052) (9)


120588119892(1199051 1199052) = [119877

119903

(1199051) sdot 119877119903

(1199052) sdot 120588119877 (1199051 1199052) + 119878119903

(1199051) sdot 119878119903

(1199052)

sdot 120588119878(1199051 1199052) minus 119877

119903

(1199051) sdot 119878119903

(1199052) sdot 120588119877119878 (1199051 1199052) minus 119877119903

(1199052) sdot 119878119903

(1199051)

sdot 120588119877119878(1199052 1199051)]

sdot1

radic[(119877119903 (1199051))2+ (119878119903 (1199051))

2minus 2119877119903 (1199051) sdot 119878

119903 (1199051) sdot 120588119877119878 (1199051 1199051)]

sdot1

radic[(119877119903 (1199052))2+ (119878119903 (1199052))

2minus 2119877119903 (1199052) sdot 119878

119903 (1199052) sdot 120588119877119878 (1199052 1199052)]

(10)






119875119891(119879) = Pr exist119905 isin [0 119879] 119892 (119905X (119905)) le 0 (11)

or

119877119904(119879) = 1minus119875

119891(119879)




119891(119879) or 119877





119875119891(119879) = Pr (119892 (0X (0)) lt 0) cup (119873+

(0 119879) gt 0) (13)


that is

119875119891(119879) le 119875

119891(0) +int

119879

0] (119905) 119889119905 (14)


T

0

g(t)


g(t) = g(X(t))




] (119905)

= limΔ119905rarr 0

Pr (119892 (119905X (119905)) gt 0) cap (119892 (119905 + Δ119905X (119905 + Δ119905)) lt 0)Δ119905

(15)


] (119905)


Δ119905

(16)





119892 (0) 119892 (Δ119905) 119892 (2Δ119905) 119892 (119873Δ119905)

119873 =119879


(17)


119894

119892(119894Δ119905) gt 0 cap 119892((119894 + 1)Δ119905) lt 0


] (119894Δ119905) asympPos 119864

119894

Δ119905

=Pos (119892 (119894Δ119905) gt 0) cap (119892 ((119894 + 1) Δ119905) lt 0)

Δ119905

(18)


119894



=119860119894


total

(19)


119877119904(119879) = 1minus119875

119891(119879)


sum

119894=1(] (119894Δ119905) sdot Δ119905))

= 1minus(Pos (0) +119873

sum

119894=1(Pos 119864

119894))

= 1minus(Pos (0) +119873

sum

119894=1(119860


total))

(20)



(1199032radic2)119892119903

(119905) lt 119892119888

(119905) lt 119892119903


119894is

equivalent to

119864119894 119892 (119894Δ119905) gt 0cap119892 ((119894 + 1) Δ119905) lt 0 997904rArr 119892

119888

(119894Δ119905)

+ 119892119903

(119894Δ119905) sdot 1198801 gt 0cap119892119888

((119894 + 1) Δ119905)

+ 119892119903


(119894Δ119905)

119892119903 (119894Δ119905)

cap1198802 ltminus119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(21)



g((i + 1)Δt)

g((i + 1)Δt)

gc((i + 1)Δt)


gc(iΔt)

gr(iΔt)

g(iΔt) g(iΔt)

gr((i + 1)Δt)


gr((i + 1)Δt)

U2

r1

0

r2

U1

U1 = minusgc(iΔt)

gr(iΔt)

(a) (b)



119864119894997904rArr 1198801 gtminus

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap1198802 ltminus

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


(119894Δ119905)

119892119903 (119894Δ119905)


((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

997904rArr 1205772 lt119903111990321205771 +

radic21199032

sdot119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap 1205772 ltminus

119903111990321205771

minusradic21199032

sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(22)








(119894Δ119905)

119892119903 (119894Δ119905)= 0


(23)


(11990321 + 119903

22) sdot 120577


119892119888

(119894Δ119905)

119892119903 (119894Δ119905)sdot 1205772 + 2(

119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2

minus 11990321 = 0

(24)

1ζ

2ζ

O

A

D

C

B

E

R = 1d



1205772 119860=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)


119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(25)

1205772 119861=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)


119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(26)




obtained


1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0


(27)

we arrive at

(11990321 + 119903

22) sdot 120577

22 + 2radic21199032 sdot

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

sdot 1205772

+ 2(119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2minus 119903

21 = 0

(28)


1205772 119862=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(29)

1205772 119863=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(30)



determined




(119894Δ119905)

119892119903 (119894Δ119905)= 0

1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0(31)

Then we have1205771 119864

=minusradic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) + 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))

21199031

1205772 119864

=

radic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) minus 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))21199032

(32)





ang119860119874119863 = tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

) (33)

The area 11987810067040 119860119862119861119863

equals

11987810067040 119860119862119861119863

=12ang119860119874119863 sdot 119877

2=12ang119860119874119863 sdot 12

=12(tanminus1 (

10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

))

(34)



and 119878Δ119863119874119864



119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119860

1205771 119864

0 1205772 1198601205772 119864

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119860

sdot 1205772 119864minus 1205771 119864

sdot 1205772 119860)

(35)

Similarly 119878Δ119863119874119864

is

119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119864

1205771 119863

0 1205772 1198641205772 119863

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119864

sdot 1205772 119863minus 1205771 119863

sdot 1205772 119864)

(36)






namely

119878∙

119860119862119861= 119878

10067040 119860119862119861minus 119878

Δ119860119874119861

11987810067040 119860119862119861

=12ang119860119874119861 sdot 119877

2=12ang119860119874119861 sdot 12


119878Δ119860119874119861

=12119860119861 sdot 119889 =


(37)


119888

(119894Δ119905)119892119903

(119894Δ119905)



Solution of

+

Begin

End

NoYes





Eq (20)

i lt N


i = i + 1

solving PosEi

Posf(0)

Pf(T) and Rs(T)

N

sumi=1

(PosEi)




Ntotal

(i)

(ii)(iii)

(iv)

(v)



No

No

No

Yes

Yes

Yes

Yes

No


N = TΔt

No

No

i = i + 1

j = 0

j = 0

i = i + 1

i = 1

i gt n



xi(NΔt)

xi isin Ωi


g(jΔt) gt 0

j lt N

j = j + 1



Ngenerated




Pos 119864119894 =

119860119894


total=119878119861119863119864

120587

=119878

10067040 119860119862119861119863minus 119878

Δ119860119874119864minus 119878

Δ119863119874119864minus 119878

∙

119860119862119861

120587

(38)








119894

119883119894(0) 119883

119894(Δ119905) 119883

119894(119873Δ119905)119873 = 119879Δ119905 in sequence


119894(119905) 119883119903

119894(119905)

and 120588119883119894



Ωi

(Xi minusXci )

119879

sdot(

11986211 11986212 1198621119873+1

11986221 11986222 1198622119873+1

119862119873+11 119862

119873+12 119862119873+1119873+1

)

minus1

(Xi minusXci )

le 1

119894 = 1 2 119899

(39)

where 119862119897119904= Cov

119883119894



119894(Δ119905)












d1

d2

F1(t)F2(t)

F3(t)l







119892 (119905) = 119892 (1198651 (119905) 1198652 (119905) 1198653 (119905) 119872criteria (119905))


sdot 1198653 (119905) 119905 isin [0 119879]

(40)










2(119905) 119865

3(119905)119872criteria(119905)

119865119888

1(119905) =

1

2sin(4120587119905 + 120587

4) +

5

2

119865119903

1(119905) = 119865

119888

1(119905) times 6120572

(unit N)

119865119888

2(119905) = minus cos(8120587119905 minus 120587

3) + 5

119865119903

2(119905) = 119865

119888

2(119905) times 5120572

(unit N)

119865119888

3(119905) = minus

3

4cos(6120587119905 + 2120587

5) +

3

2

119865119903

3(119905) = 119865

119888

3(119905) times 3120572

(unit N)

119872119888


5119905

119872119903

criteria (119905) =120572119905

2

100


1(119905) 119865

2(119905) 119865

3(119905)119872criteria(119905)

1205881198651

(1199051 119905

2) = 119890


(1199051 119905

2) = 119890



(1199051 119905

2) = 119890


(1199051 119905

2) =

1

2(1 + cos (6119896 sdot 1003816100381610038161003816t1 minus t2

1003816100381610038161003816))


119896120572 120572 = 10 120572 = 12 120572 = 14 120572 = 16 120572 = 18 120572 = 20119896 = 1 1 0986 09426 08931 08259 07595119896 = 2 1 09857 09424 08881 08155 07425119896 = 3 1 09854 09423 08821 08027 07249119896 = 4 1 09853 09419 08741 07915 07085119896 = 5 1 09852 09406 08696 07844 06972

10 2018161412070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

k = 1 k = 2k = 3

k = 4k = 5

120572 (T = 10 years)








1 2 3 4 5068

070

072

074

076

078

080

Tim

e-de

pend

ent r

eliab

ility





119904(119879) is widening










2(119905) 119865

3(119905) 119877


119865119888

1(119905) = 119865

119888

2(119905) = 2 sin(3120587119905 + 120587

3) + 4448

119865119903

1(119905) = 119865

119903

2(119905) = 119865

119888

1(119905) times 3120572

(unit KN)

119865119888

3(119905) = 7 cos (4120587119905 minus 120587

5) + 17792

119865119903

3(119905) = 119865

119888

3(119905) times 2120572

(unit KN)

119877119888

119879criteria(119905) = 375 minus15119905

8119877119903

119879criteria(119905) =120572119905

2

36

119877119888

119862criteria(119905) = minus100 +3119905

4119877119903

119862criteria(119905) =120572119905

2

32


1(119905) 119865

2(119905) 119865

3(119905) 119877


1205881198651

(1199051 119905

2) = 119890

minus(5|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198651(119905) and 119865

2(119905)120588

1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 120588


(1199051 119905

2) = 0

1

246

35

l l

X

Y

43

108

65

97

21

l

F1(t) F2(t)

F3(t)


119877119879criteria(119905) and 119877



119905 isin [0 119879]

(41)

1198922 (119905) = 119892 (119878119862max (119905) 119877119862criteria (119905))

=1003816100381610038161003816119877119862criteria (119905)

1003816100381610038161003816

minus1003816100381610038161003816119878119862max (1198651 (119905) 1198652 (119905) 1198653 (119905))

1003816100381610038161003816

119905 isin [0 119879]

(42)









12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 09096 0873 08329 07904 07369 0671412058811986511198652

= minus01 09085 08704 08319 07873 07327 0664712058811986511198652

= 0 0906 08684 08302 07842 07284 0663512058811986511198652

= 01 09053 08675 08296 07821 07272 065912058811986511198652

= 02 09047 08669 08276 07805 07238 06579


12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 1 09947 09581 08928 07899 0652112058811986511198652

= minus01 099941 09817 0929 08385 07075 0565812058811986511198652

= 0 09954 09603 08888 07762 0633 048212058811986511198652

= 01 09859 0935 08405 07057 05589 0402812058811986511198652

= 02 09706 09003 07858 06381 04884 03216


11986511198652is




(3) 12058811986511198652







15 2019181716065

070

075

080

085

090


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)


15 2019181716060

065

070

075

080

085

090

095

Tim

e-de

pend

ent r

eliab

ility




15 2019181716

04

05

06

07

08

09

10

Tim

e-de

pend

ent r

eliab

ility





119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

(a) (b)

0

02

2

X(t2)

X(t2)

Xc(t2)

X(t2)


Xr(t1)

Xr(t2)

U1

U2





Cov119883(1199051 1199052) = Cov (1198801 1198802) sdot 119883

119903

(1199051) sdot 119883119903

(1199052)

= (1199032119898minus 1) sdot 119883119903

(1199051) sdot 119883119903

(1199052)

0 le 119903119898le radic2

(3)




120588119883(1199051 1199052) =

Cov119883(1199051 1199052)

radic119863119883(1199051) sdot radic119863119883

(1199052)=

Cov (1198801 1198802)


1198802

= 12058811988011198802

= 1199032119898minus 1

(4)

where 1198631198801

and 1198631198802


1198801= 119863

1198802=



119883(1199051 1199052)| le 1 and 120588

119883(119905 119905) = 1








120588119883119884(1199051 1199052) =

Cov119883119884(1199051 1199052)

radic119863119883(1199051) sdot radic119863119884

(1199052)= 119903



(5)






0

2

2 0

2

2 U1

U2

5∘120579 = minus4U1

U2

120579 = 45∘

r1

r1

r2

r2

rm = r1

rm = r2

(a) (b)


120588X(t1 t2) = 05

120588X(t1 t2) = 03

120588X(t1 t2) = 1

120588X(t1 t2) = 0

120588X(t1 t2) = 08

U1

U2

0

120588X(t1 t2) = minus05

120588X(t1 t2) = minus03

120588X(t1 t2) = minus1

120588X(t1 t2) = 0

120588X(t1 t2) = minus08

U1

U2

0





119892 (119905) = 119892 (119905 a (119905) X (119905)) = 1198860 (119905) +119899

sum

119894=1(119886

119894(119905) sdot 119883

119894(119905)) (6)

where X(119905) = (1198831(119905) 1198832(119905) 119883119899(119905))





119892119888

(119905) = 119892119888

(119905 a (119905) X (119905)) = 1198860 (119905) +119899

sum

119894=1(119886

119894(119905) sdot 119883

119888

119894(119905))

119892119903

(119905) = radic119863119892(119905) = 119892

119903

(119905 a (119905) X (119905)) =radic

119899

sum

119894=1(119886

119894(119905) sdot 119883

119903

119894(119905))

2+

119899

sum

119894=1

119899

sum

119895=1119895 =119894

120588119883119894119883119895

(119905 119905) sdot 119886119894(119905) sdot 119886

119895(119905) sdot 119883

119903

119894(119905) sdot 119883

119903

119895(119905)

(7)

where Xc(119905) = (119883

119888

1(119905) 119883119888

2(119905) 119883119888

119899(119905))

119879 and Xr(119905) = (119883

119903

1(119905)

119883119903

2(119905) 119883119903

119899(119905))


119883119894119883119895


119894(119905) and119883

119895(119905)



119894(1199051) and119883119894


119894(1199051) and 119883119895


Cov119892(1199051 1199052) =

119899

sum

119894=1120588119883119894

(1199051 1199052) sdot 119886119894 (1199051) sdot 119886119894 (1199052) sdot 119883119903

119894(1199051)

sdot 119883119903

119894(1199052) +

119899

sum

119894=1

119899

sum

119895=1119895 =119894

120588119883119894119883119895

(1199051 1199052) sdot 119886119894 (1199051) sdot 119886119895 (1199052)

sdot 119883119903

119894(1199051) sdot 119883

119903

119895(1199052)

(8)


of 119883119894(1199051) and 119883119894

(1199052) and 120588119883119894119883119895


119894(1199051) and119883119895


120588119892(1199051 1199052) =

Cov119892(1199051 1199052)

radic119863119892(1199051) sdot radic119863119892

(1199052) (9)


120588119892(1199051 1199052) = [119877

119903

(1199051) sdot 119877119903

(1199052) sdot 120588119877 (1199051 1199052) + 119878119903

(1199051) sdot 119878119903

(1199052)

sdot 120588119878(1199051 1199052) minus 119877

119903

(1199051) sdot 119878119903

(1199052) sdot 120588119877119878 (1199051 1199052) minus 119877119903

(1199052) sdot 119878119903

(1199051)

sdot 120588119877119878(1199052 1199051)]

sdot1

radic[(119877119903 (1199051))2+ (119878119903 (1199051))

2minus 2119877119903 (1199051) sdot 119878

119903 (1199051) sdot 120588119877119878 (1199051 1199051)]

sdot1

radic[(119877119903 (1199052))2+ (119878119903 (1199052))

2minus 2119877119903 (1199052) sdot 119878

119903 (1199052) sdot 120588119877119878 (1199052 1199052)]

(10)






119875119891(119879) = Pr exist119905 isin [0 119879] 119892 (119905X (119905)) le 0 (11)

or

119877119904(119879) = 1minus119875

119891(119879)




119891(119879) or 119877





119875119891(119879) = Pr (119892 (0X (0)) lt 0) cup (119873+

(0 119879) gt 0) (13)


that is

119875119891(119879) le 119875

119891(0) +int

119879

0] (119905) 119889119905 (14)


T

0

g(t)


g(t) = g(X(t))




] (119905)

= limΔ119905rarr 0

Pr (119892 (119905X (119905)) gt 0) cap (119892 (119905 + Δ119905X (119905 + Δ119905)) lt 0)Δ119905

(15)


] (119905)


Δ119905

(16)





119892 (0) 119892 (Δ119905) 119892 (2Δ119905) 119892 (119873Δ119905)

119873 =119879


(17)


119894

119892(119894Δ119905) gt 0 cap 119892((119894 + 1)Δ119905) lt 0


] (119894Δ119905) asympPos 119864

119894

Δ119905

=Pos (119892 (119894Δ119905) gt 0) cap (119892 ((119894 + 1) Δ119905) lt 0)

Δ119905

(18)


119894



=119860119894


total

(19)


119877119904(119879) = 1minus119875

119891(119879)


sum

119894=1(] (119894Δ119905) sdot Δ119905))

= 1minus(Pos (0) +119873

sum

119894=1(Pos 119864

119894))

= 1minus(Pos (0) +119873

sum

119894=1(119860


total))

(20)



(1199032radic2)119892119903

(119905) lt 119892119888

(119905) lt 119892119903


119894is

equivalent to

119864119894 119892 (119894Δ119905) gt 0cap119892 ((119894 + 1) Δ119905) lt 0 997904rArr 119892

119888

(119894Δ119905)

+ 119892119903

(119894Δ119905) sdot 1198801 gt 0cap119892119888

((119894 + 1) Δ119905)

+ 119892119903


(119894Δ119905)

119892119903 (119894Δ119905)

cap1198802 ltminus119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(21)



g((i + 1)Δt)

g((i + 1)Δt)

gc((i + 1)Δt)


gc(iΔt)

gr(iΔt)

g(iΔt) g(iΔt)

gr((i + 1)Δt)


gr((i + 1)Δt)

U2

r1

0

r2

U1

U1 = minusgc(iΔt)

gr(iΔt)

(a) (b)



119864119894997904rArr 1198801 gtminus

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap1198802 ltminus

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


(119894Δ119905)

119892119903 (119894Δ119905)


((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

997904rArr 1205772 lt119903111990321205771 +

radic21199032

sdot119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap 1205772 ltminus

119903111990321205771

minusradic21199032

sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(22)








(119894Δ119905)

119892119903 (119894Δ119905)= 0


(23)


(11990321 + 119903

22) sdot 120577


119892119888

(119894Δ119905)

119892119903 (119894Δ119905)sdot 1205772 + 2(

119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2

minus 11990321 = 0

(24)

1ζ

2ζ

O

A

D

C

B

E

R = 1d



1205772 119860=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)


119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(25)

1205772 119861=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)


119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(26)




obtained


1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0


(27)

we arrive at

(11990321 + 119903

22) sdot 120577

22 + 2radic21199032 sdot

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

sdot 1205772

+ 2(119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2minus 119903

21 = 0

(28)


1205772 119862=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(29)

1205772 119863=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(30)



determined




(119894Δ119905)

119892119903 (119894Δ119905)= 0

1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0(31)

Then we have1205771 119864

=minusradic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) + 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))

21199031

1205772 119864

=

radic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) minus 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))21199032

(32)





ang119860119874119863 = tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

) (33)

The area 11987810067040 119860119862119861119863

equals

11987810067040 119860119862119861119863

=12ang119860119874119863 sdot 119877

2=12ang119860119874119863 sdot 12

=12(tanminus1 (

10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

))

(34)



and 119878Δ119863119874119864



119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119860

1205771 119864

0 1205772 1198601205772 119864

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119860

sdot 1205772 119864minus 1205771 119864

sdot 1205772 119860)

(35)

Similarly 119878Δ119863119874119864

is

119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119864

1205771 119863

0 1205772 1198641205772 119863

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119864

sdot 1205772 119863minus 1205771 119863

sdot 1205772 119864)

(36)






namely

119878∙

119860119862119861= 119878

10067040 119860119862119861minus 119878

Δ119860119874119861

11987810067040 119860119862119861

=12ang119860119874119861 sdot 119877

2=12ang119860119874119861 sdot 12


119878Δ119860119874119861

=12119860119861 sdot 119889 =


(37)


119888

(119894Δ119905)119892119903

(119894Δ119905)



Solution of

+

Begin

End

NoYes





Eq (20)

i lt N


i = i + 1

solving PosEi

Posf(0)

Pf(T) and Rs(T)

N

sumi=1

(PosEi)




Ntotal

(i)

(ii)(iii)

(iv)

(v)



No

No

No

Yes

Yes

Yes

Yes

No


N = TΔt

No

No

i = i + 1

j = 0

j = 0

i = i + 1

i = 1

i gt n



xi(NΔt)

xi isin Ωi


g(jΔt) gt 0

j lt N

j = j + 1



Ngenerated




Pos 119864119894 =

119860119894


total=119878119861119863119864

120587

=119878

10067040 119860119862119861119863minus 119878

Δ119860119874119864minus 119878

Δ119863119874119864minus 119878

∙

119860119862119861

120587

(38)








119894

119883119894(0) 119883

119894(Δ119905) 119883

119894(119873Δ119905)119873 = 119879Δ119905 in sequence


119894(119905) 119883119903

119894(119905)

and 120588119883119894



Ωi

(Xi minusXci )

119879

sdot(

11986211 11986212 1198621119873+1

11986221 11986222 1198622119873+1

119862119873+11 119862

119873+12 119862119873+1119873+1

)

minus1

(Xi minusXci )

le 1

119894 = 1 2 119899

(39)

where 119862119897119904= Cov

119883119894



119894(Δ119905)












d1

d2

F1(t)F2(t)

F3(t)l







119892 (119905) = 119892 (1198651 (119905) 1198652 (119905) 1198653 (119905) 119872criteria (119905))


sdot 1198653 (119905) 119905 isin [0 119879]

(40)










2(119905) 119865

3(119905)119872criteria(119905)

119865119888

1(119905) =

1

2sin(4120587119905 + 120587

4) +

5

2

119865119903

1(119905) = 119865

119888

1(119905) times 6120572

(unit N)

119865119888

2(119905) = minus cos(8120587119905 minus 120587

3) + 5

119865119903

2(119905) = 119865

119888

2(119905) times 5120572

(unit N)

119865119888

3(119905) = minus

3

4cos(6120587119905 + 2120587

5) +

3

2

119865119903

3(119905) = 119865

119888

3(119905) times 3120572

(unit N)

119872119888


5119905

119872119903

criteria (119905) =120572119905

2

100


1(119905) 119865

2(119905) 119865

3(119905)119872criteria(119905)

1205881198651

(1199051 119905

2) = 119890


(1199051 119905

2) = 119890



(1199051 119905

2) = 119890


(1199051 119905

2) =

1

2(1 + cos (6119896 sdot 1003816100381610038161003816t1 minus t2

1003816100381610038161003816))


119896120572 120572 = 10 120572 = 12 120572 = 14 120572 = 16 120572 = 18 120572 = 20119896 = 1 1 0986 09426 08931 08259 07595119896 = 2 1 09857 09424 08881 08155 07425119896 = 3 1 09854 09423 08821 08027 07249119896 = 4 1 09853 09419 08741 07915 07085119896 = 5 1 09852 09406 08696 07844 06972

10 2018161412070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

k = 1 k = 2k = 3

k = 4k = 5

120572 (T = 10 years)








1 2 3 4 5068

070

072

074

076

078

080

Tim

e-de

pend

ent r

eliab

ility





119904(119879) is widening










2(119905) 119865

3(119905) 119877


119865119888

1(119905) = 119865

119888

2(119905) = 2 sin(3120587119905 + 120587

3) + 4448

119865119903

1(119905) = 119865

119903

2(119905) = 119865

119888

1(119905) times 3120572

(unit KN)

119865119888

3(119905) = 7 cos (4120587119905 minus 120587

5) + 17792

119865119903

3(119905) = 119865

119888

3(119905) times 2120572

(unit KN)

119877119888

119879criteria(119905) = 375 minus15119905

8119877119903

119879criteria(119905) =120572119905

2

36

119877119888

119862criteria(119905) = minus100 +3119905

4119877119903

119862criteria(119905) =120572119905

2

32


1(119905) 119865

2(119905) 119865

3(119905) 119877


1205881198651

(1199051 119905

2) = 119890

minus(5|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198651(119905) and 119865

2(119905)120588

1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 120588


(1199051 119905

2) = 0

1

246

35

l l

X

Y

43

108

65

97

21

l

F1(t) F2(t)

F3(t)


119877119879criteria(119905) and 119877



119905 isin [0 119879]

(41)

1198922 (119905) = 119892 (119878119862max (119905) 119877119862criteria (119905))

=1003816100381610038161003816119877119862criteria (119905)

1003816100381610038161003816

minus1003816100381610038161003816119878119862max (1198651 (119905) 1198652 (119905) 1198653 (119905))

1003816100381610038161003816

119905 isin [0 119879]

(42)









12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 09096 0873 08329 07904 07369 0671412058811986511198652

= minus01 09085 08704 08319 07873 07327 0664712058811986511198652

= 0 0906 08684 08302 07842 07284 0663512058811986511198652

= 01 09053 08675 08296 07821 07272 065912058811986511198652

= 02 09047 08669 08276 07805 07238 06579


12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 1 09947 09581 08928 07899 0652112058811986511198652

= minus01 099941 09817 0929 08385 07075 0565812058811986511198652

= 0 09954 09603 08888 07762 0633 048212058811986511198652

= 01 09859 0935 08405 07057 05589 0402812058811986511198652

= 02 09706 09003 07858 06381 04884 03216


11986511198652is




(3) 12058811986511198652







15 2019181716065

070

075

080

085

090


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)


15 2019181716060

065

070

075

080

085

090

095

Tim

e-de

pend

ent r

eliab

ility




15 2019181716

04

05

06

07

08

09

10

Tim

e-de

pend

ent r

eliab

ility





119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2

2 0

2

2 U1

U2

5∘120579 = minus4U1

U2

120579 = 45∘

r1

r1

r2

r2

rm = r1

rm = r2

(a) (b)


120588X(t1 t2) = 05

120588X(t1 t2) = 03

120588X(t1 t2) = 1

120588X(t1 t2) = 0

120588X(t1 t2) = 08

U1

U2

0

120588X(t1 t2) = minus05

120588X(t1 t2) = minus03

120588X(t1 t2) = minus1

120588X(t1 t2) = 0

120588X(t1 t2) = minus08

U1

U2

0





119892 (119905) = 119892 (119905 a (119905) X (119905)) = 1198860 (119905) +119899

sum

119894=1(119886

119894(119905) sdot 119883

119894(119905)) (6)

where X(119905) = (1198831(119905) 1198832(119905) 119883119899(119905))





119892119888

(119905) = 119892119888

(119905 a (119905) X (119905)) = 1198860 (119905) +119899

sum

119894=1(119886

119894(119905) sdot 119883

119888

119894(119905))

119892119903

(119905) = radic119863119892(119905) = 119892

119903

(119905 a (119905) X (119905)) =radic

119899

sum

119894=1(119886

119894(119905) sdot 119883

119903

119894(119905))

2+

119899

sum

119894=1

119899

sum

119895=1119895 =119894

120588119883119894119883119895

(119905 119905) sdot 119886119894(119905) sdot 119886

119895(119905) sdot 119883

119903

119894(119905) sdot 119883

119903

119895(119905)

(7)

where Xc(119905) = (119883

119888

1(119905) 119883119888

2(119905) 119883119888

119899(119905))

119879 and Xr(119905) = (119883

119903

1(119905)

119883119903

2(119905) 119883119903

119899(119905))


119883119894119883119895


119894(119905) and119883

119895(119905)



119894(1199051) and119883119894


119894(1199051) and 119883119895


Cov119892(1199051 1199052) =

119899

sum

119894=1120588119883119894

(1199051 1199052) sdot 119886119894 (1199051) sdot 119886119894 (1199052) sdot 119883119903

119894(1199051)

sdot 119883119903

119894(1199052) +

119899

sum

119894=1

119899

sum

119895=1119895 =119894

120588119883119894119883119895

(1199051 1199052) sdot 119886119894 (1199051) sdot 119886119895 (1199052)

sdot 119883119903

119894(1199051) sdot 119883

119903

119895(1199052)

(8)


of 119883119894(1199051) and 119883119894

(1199052) and 120588119883119894119883119895


119894(1199051) and119883119895


120588119892(1199051 1199052) =

Cov119892(1199051 1199052)

radic119863119892(1199051) sdot radic119863119892

(1199052) (9)


120588119892(1199051 1199052) = [119877

119903

(1199051) sdot 119877119903

(1199052) sdot 120588119877 (1199051 1199052) + 119878119903

(1199051) sdot 119878119903

(1199052)

sdot 120588119878(1199051 1199052) minus 119877

119903

(1199051) sdot 119878119903

(1199052) sdot 120588119877119878 (1199051 1199052) minus 119877119903

(1199052) sdot 119878119903

(1199051)

sdot 120588119877119878(1199052 1199051)]

sdot1

radic[(119877119903 (1199051))2+ (119878119903 (1199051))

2minus 2119877119903 (1199051) sdot 119878

119903 (1199051) sdot 120588119877119878 (1199051 1199051)]

sdot1

radic[(119877119903 (1199052))2+ (119878119903 (1199052))

2minus 2119877119903 (1199052) sdot 119878

119903 (1199052) sdot 120588119877119878 (1199052 1199052)]

(10)






119875119891(119879) = Pr exist119905 isin [0 119879] 119892 (119905X (119905)) le 0 (11)

or

119877119904(119879) = 1minus119875

119891(119879)




119891(119879) or 119877





119875119891(119879) = Pr (119892 (0X (0)) lt 0) cup (119873+

(0 119879) gt 0) (13)


that is

119875119891(119879) le 119875

119891(0) +int

119879

0] (119905) 119889119905 (14)


T

0

g(t)


g(t) = g(X(t))




] (119905)

= limΔ119905rarr 0

Pr (119892 (119905X (119905)) gt 0) cap (119892 (119905 + Δ119905X (119905 + Δ119905)) lt 0)Δ119905

(15)


] (119905)


Δ119905

(16)





119892 (0) 119892 (Δ119905) 119892 (2Δ119905) 119892 (119873Δ119905)

119873 =119879


(17)


119894

119892(119894Δ119905) gt 0 cap 119892((119894 + 1)Δ119905) lt 0


] (119894Δ119905) asympPos 119864

119894

Δ119905

=Pos (119892 (119894Δ119905) gt 0) cap (119892 ((119894 + 1) Δ119905) lt 0)

Δ119905

(18)


119894



=119860119894


total

(19)


119877119904(119879) = 1minus119875

119891(119879)


sum

119894=1(] (119894Δ119905) sdot Δ119905))

= 1minus(Pos (0) +119873

sum

119894=1(Pos 119864

119894))

= 1minus(Pos (0) +119873

sum

119894=1(119860


total))

(20)



(1199032radic2)119892119903

(119905) lt 119892119888

(119905) lt 119892119903


119894is

equivalent to

119864119894 119892 (119894Δ119905) gt 0cap119892 ((119894 + 1) Δ119905) lt 0 997904rArr 119892

119888

(119894Δ119905)

+ 119892119903

(119894Δ119905) sdot 1198801 gt 0cap119892119888

((119894 + 1) Δ119905)

+ 119892119903


(119894Δ119905)

119892119903 (119894Δ119905)

cap1198802 ltminus119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(21)



g((i + 1)Δt)

g((i + 1)Δt)

gc((i + 1)Δt)


gc(iΔt)

gr(iΔt)

g(iΔt) g(iΔt)

gr((i + 1)Δt)


gr((i + 1)Δt)

U2

r1

0

r2

U1

U1 = minusgc(iΔt)

gr(iΔt)

(a) (b)



119864119894997904rArr 1198801 gtminus

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap1198802 ltminus

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


(119894Δ119905)

119892119903 (119894Δ119905)


((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

997904rArr 1205772 lt119903111990321205771 +

radic21199032

sdot119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap 1205772 ltminus

119903111990321205771

minusradic21199032

sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(22)








(119894Δ119905)

119892119903 (119894Δ119905)= 0


(23)


(11990321 + 119903

22) sdot 120577


119892119888

(119894Δ119905)

119892119903 (119894Δ119905)sdot 1205772 + 2(

119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2

minus 11990321 = 0

(24)

1ζ

2ζ

O

A

D

C

B

E

R = 1d



1205772 119860=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)


119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(25)

1205772 119861=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)


119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(26)




obtained


1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0


(27)

we arrive at

(11990321 + 119903

22) sdot 120577

22 + 2radic21199032 sdot

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

sdot 1205772

+ 2(119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2minus 119903

21 = 0

(28)


1205772 119862=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(29)

1205772 119863=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(30)



determined




(119894Δ119905)

119892119903 (119894Δ119905)= 0

1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0(31)

Then we have1205771 119864

=minusradic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) + 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))

21199031

1205772 119864

=

radic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) minus 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))21199032

(32)





ang119860119874119863 = tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

) (33)

The area 11987810067040 119860119862119861119863

equals

11987810067040 119860119862119861119863

=12ang119860119874119863 sdot 119877

2=12ang119860119874119863 sdot 12

=12(tanminus1 (

10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

))

(34)



and 119878Δ119863119874119864



119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119860

1205771 119864

0 1205772 1198601205772 119864

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119860

sdot 1205772 119864minus 1205771 119864

sdot 1205772 119860)

(35)

Similarly 119878Δ119863119874119864

is

119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119864

1205771 119863

0 1205772 1198641205772 119863

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119864

sdot 1205772 119863minus 1205771 119863

sdot 1205772 119864)

(36)






namely

119878∙

119860119862119861= 119878

10067040 119860119862119861minus 119878

Δ119860119874119861

11987810067040 119860119862119861

=12ang119860119874119861 sdot 119877

2=12ang119860119874119861 sdot 12


119878Δ119860119874119861

=12119860119861 sdot 119889 =


(37)


119888

(119894Δ119905)119892119903

(119894Δ119905)



Solution of

+

Begin

End

NoYes





Eq (20)

i lt N


i = i + 1

solving PosEi

Posf(0)

Pf(T) and Rs(T)

N

sumi=1

(PosEi)




Ntotal

(i)

(ii)(iii)

(iv)

(v)



No

No

No

Yes

Yes

Yes

Yes

No


N = TΔt

No

No

i = i + 1

j = 0

j = 0

i = i + 1

i = 1

i gt n



xi(NΔt)

xi isin Ωi


g(jΔt) gt 0

j lt N

j = j + 1



Ngenerated




Pos 119864119894 =

119860119894


total=119878119861119863119864

120587

=119878

10067040 119860119862119861119863minus 119878

Δ119860119874119864minus 119878

Δ119863119874119864minus 119878

∙

119860119862119861

120587

(38)








119894

119883119894(0) 119883

119894(Δ119905) 119883

119894(119873Δ119905)119873 = 119879Δ119905 in sequence


119894(119905) 119883119903

119894(119905)

and 120588119883119894



Ωi

(Xi minusXci )

119879

sdot(

11986211 11986212 1198621119873+1

11986221 11986222 1198622119873+1

119862119873+11 119862

119873+12 119862119873+1119873+1

)

minus1

(Xi minusXci )

le 1

119894 = 1 2 119899

(39)

where 119862119897119904= Cov

119883119894



119894(Δ119905)












d1

d2

F1(t)F2(t)

F3(t)l







119892 (119905) = 119892 (1198651 (119905) 1198652 (119905) 1198653 (119905) 119872criteria (119905))


sdot 1198653 (119905) 119905 isin [0 119879]

(40)










2(119905) 119865

3(119905)119872criteria(119905)

119865119888

1(119905) =

1

2sin(4120587119905 + 120587

4) +

5

2

119865119903

1(119905) = 119865

119888

1(119905) times 6120572

(unit N)

119865119888

2(119905) = minus cos(8120587119905 minus 120587

3) + 5

119865119903

2(119905) = 119865

119888

2(119905) times 5120572

(unit N)

119865119888

3(119905) = minus

3

4cos(6120587119905 + 2120587

5) +

3

2

119865119903

3(119905) = 119865

119888

3(119905) times 3120572

(unit N)

119872119888


5119905

119872119903

criteria (119905) =120572119905

2

100


1(119905) 119865

2(119905) 119865

3(119905)119872criteria(119905)

1205881198651

(1199051 119905

2) = 119890


(1199051 119905

2) = 119890



(1199051 119905

2) = 119890


(1199051 119905

2) =

1

2(1 + cos (6119896 sdot 1003816100381610038161003816t1 minus t2

1003816100381610038161003816))


119896120572 120572 = 10 120572 = 12 120572 = 14 120572 = 16 120572 = 18 120572 = 20119896 = 1 1 0986 09426 08931 08259 07595119896 = 2 1 09857 09424 08881 08155 07425119896 = 3 1 09854 09423 08821 08027 07249119896 = 4 1 09853 09419 08741 07915 07085119896 = 5 1 09852 09406 08696 07844 06972

10 2018161412070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

k = 1 k = 2k = 3

k = 4k = 5

120572 (T = 10 years)








1 2 3 4 5068

070

072

074

076

078

080

Tim

e-de

pend

ent r

eliab

ility





119904(119879) is widening










2(119905) 119865

3(119905) 119877


119865119888

1(119905) = 119865

119888

2(119905) = 2 sin(3120587119905 + 120587

3) + 4448

119865119903

1(119905) = 119865

119903

2(119905) = 119865

119888

1(119905) times 3120572

(unit KN)

119865119888

3(119905) = 7 cos (4120587119905 minus 120587

5) + 17792

119865119903

3(119905) = 119865

119888

3(119905) times 2120572

(unit KN)

119877119888

119879criteria(119905) = 375 minus15119905

8119877119903

119879criteria(119905) =120572119905

2

36

119877119888

119862criteria(119905) = minus100 +3119905

4119877119903

119862criteria(119905) =120572119905

2

32


1(119905) 119865

2(119905) 119865

3(119905) 119877


1205881198651

(1199051 119905

2) = 119890

minus(5|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198651(119905) and 119865

2(119905)120588

1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 120588


(1199051 119905

2) = 0

1

246

35

l l

X

Y

43

108

65

97

21

l

F1(t) F2(t)

F3(t)


119877119879criteria(119905) and 119877



119905 isin [0 119879]

(41)

1198922 (119905) = 119892 (119878119862max (119905) 119877119862criteria (119905))

=1003816100381610038161003816119877119862criteria (119905)

1003816100381610038161003816

minus1003816100381610038161003816119878119862max (1198651 (119905) 1198652 (119905) 1198653 (119905))

1003816100381610038161003816

119905 isin [0 119879]

(42)









12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 09096 0873 08329 07904 07369 0671412058811986511198652

= minus01 09085 08704 08319 07873 07327 0664712058811986511198652

= 0 0906 08684 08302 07842 07284 0663512058811986511198652

= 01 09053 08675 08296 07821 07272 065912058811986511198652

= 02 09047 08669 08276 07805 07238 06579


12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 1 09947 09581 08928 07899 0652112058811986511198652

= minus01 099941 09817 0929 08385 07075 0565812058811986511198652

= 0 09954 09603 08888 07762 0633 048212058811986511198652

= 01 09859 0935 08405 07057 05589 0402812058811986511198652

= 02 09706 09003 07858 06381 04884 03216


11986511198652is




(3) 12058811986511198652







15 2019181716065

070

075

080

085

090


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)


15 2019181716060

065

070

075

080

085

090

095

Tim

e-de

pend

ent r

eliab

ility




15 2019181716

04

05

06

07

08

09

10

Tim

e-de

pend

ent r

eliab

ility





119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119892119888

(119905) = 119892119888

(119905 a (119905) X (119905)) = 1198860 (119905) +119899

sum

119894=1(119886

119894(119905) sdot 119883

119888

119894(119905))

119892119903

(119905) = radic119863119892(119905) = 119892

119903

(119905 a (119905) X (119905)) =radic

119899

sum

119894=1(119886

119894(119905) sdot 119883

119903

119894(119905))

2+

119899

sum

119894=1

119899

sum

119895=1119895 =119894

120588119883119894119883119895

(119905 119905) sdot 119886119894(119905) sdot 119886

119895(119905) sdot 119883

119903

119894(119905) sdot 119883

119903

119895(119905)

(7)

where Xc(119905) = (119883

119888

1(119905) 119883119888

2(119905) 119883119888

119899(119905))

119879 and Xr(119905) = (119883

119903

1(119905)

119883119903

2(119905) 119883119903

119899(119905))


119883119894119883119895


119894(119905) and119883

119895(119905)



119894(1199051) and119883119894


119894(1199051) and 119883119895


Cov119892(1199051 1199052) =

119899

sum

119894=1120588119883119894

(1199051 1199052) sdot 119886119894 (1199051) sdot 119886119894 (1199052) sdot 119883119903

119894(1199051)

sdot 119883119903

119894(1199052) +

119899

sum

119894=1

119899

sum

119895=1119895 =119894

120588119883119894119883119895

(1199051 1199052) sdot 119886119894 (1199051) sdot 119886119895 (1199052)

sdot 119883119903

119894(1199051) sdot 119883

119903

119895(1199052)

(8)


of 119883119894(1199051) and 119883119894

(1199052) and 120588119883119894119883119895


119894(1199051) and119883119895


120588119892(1199051 1199052) =

Cov119892(1199051 1199052)

radic119863119892(1199051) sdot radic119863119892

(1199052) (9)


120588119892(1199051 1199052) = [119877

119903

(1199051) sdot 119877119903

(1199052) sdot 120588119877 (1199051 1199052) + 119878119903

(1199051) sdot 119878119903

(1199052)

sdot 120588119878(1199051 1199052) minus 119877

119903

(1199051) sdot 119878119903

(1199052) sdot 120588119877119878 (1199051 1199052) minus 119877119903

(1199052) sdot 119878119903

(1199051)

sdot 120588119877119878(1199052 1199051)]

sdot1

radic[(119877119903 (1199051))2+ (119878119903 (1199051))

2minus 2119877119903 (1199051) sdot 119878

119903 (1199051) sdot 120588119877119878 (1199051 1199051)]

sdot1

radic[(119877119903 (1199052))2+ (119878119903 (1199052))

2minus 2119877119903 (1199052) sdot 119878

119903 (1199052) sdot 120588119877119878 (1199052 1199052)]

(10)






119875119891(119879) = Pr exist119905 isin [0 119879] 119892 (119905X (119905)) le 0 (11)

or

119877119904(119879) = 1minus119875

119891(119879)




119891(119879) or 119877





119875119891(119879) = Pr (119892 (0X (0)) lt 0) cup (119873+

(0 119879) gt 0) (13)


that is

119875119891(119879) le 119875

119891(0) +int

119879

0] (119905) 119889119905 (14)


T

0

g(t)


g(t) = g(X(t))




] (119905)

= limΔ119905rarr 0

Pr (119892 (119905X (119905)) gt 0) cap (119892 (119905 + Δ119905X (119905 + Δ119905)) lt 0)Δ119905

(15)


] (119905)


Δ119905

(16)





119892 (0) 119892 (Δ119905) 119892 (2Δ119905) 119892 (119873Δ119905)

119873 =119879


(17)


119894

119892(119894Δ119905) gt 0 cap 119892((119894 + 1)Δ119905) lt 0


] (119894Δ119905) asympPos 119864

119894

Δ119905

=Pos (119892 (119894Δ119905) gt 0) cap (119892 ((119894 + 1) Δ119905) lt 0)

Δ119905

(18)


119894



=119860119894


total

(19)


119877119904(119879) = 1minus119875

119891(119879)


sum

119894=1(] (119894Δ119905) sdot Δ119905))

= 1minus(Pos (0) +119873

sum

119894=1(Pos 119864

119894))

= 1minus(Pos (0) +119873

sum

119894=1(119860


total))

(20)



(1199032radic2)119892119903

(119905) lt 119892119888

(119905) lt 119892119903


119894is

equivalent to

119864119894 119892 (119894Δ119905) gt 0cap119892 ((119894 + 1) Δ119905) lt 0 997904rArr 119892

119888

(119894Δ119905)

+ 119892119903

(119894Δ119905) sdot 1198801 gt 0cap119892119888

((119894 + 1) Δ119905)

+ 119892119903


(119894Δ119905)

119892119903 (119894Δ119905)

cap1198802 ltminus119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(21)



g((i + 1)Δt)

g((i + 1)Δt)

gc((i + 1)Δt)


gc(iΔt)

gr(iΔt)

g(iΔt) g(iΔt)

gr((i + 1)Δt)


gr((i + 1)Δt)

U2

r1

0

r2

U1

U1 = minusgc(iΔt)

gr(iΔt)

(a) (b)



119864119894997904rArr 1198801 gtminus

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap1198802 ltminus

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


(119894Δ119905)

119892119903 (119894Δ119905)


((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

997904rArr 1205772 lt119903111990321205771 +

radic21199032

sdot119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap 1205772 ltminus

119903111990321205771

minusradic21199032

sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(22)








(119894Δ119905)

119892119903 (119894Δ119905)= 0


(23)


(11990321 + 119903

22) sdot 120577


119892119888

(119894Δ119905)

119892119903 (119894Δ119905)sdot 1205772 + 2(

119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2

minus 11990321 = 0

(24)

1ζ

2ζ

O

A

D

C

B

E

R = 1d



1205772 119860=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)


119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(25)

1205772 119861=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)


119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(26)




obtained


1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0


(27)

we arrive at

(11990321 + 119903

22) sdot 120577

22 + 2radic21199032 sdot

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

sdot 1205772

+ 2(119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2minus 119903

21 = 0

(28)


1205772 119862=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(29)

1205772 119863=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(30)



determined




(119894Δ119905)

119892119903 (119894Δ119905)= 0

1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0(31)

Then we have1205771 119864

=minusradic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) + 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))

21199031

1205772 119864

=

radic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) minus 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))21199032

(32)





ang119860119874119863 = tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

) (33)

The area 11987810067040 119860119862119861119863

equals

11987810067040 119860119862119861119863

=12ang119860119874119863 sdot 119877

2=12ang119860119874119863 sdot 12

=12(tanminus1 (

10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

))

(34)



and 119878Δ119863119874119864



119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119860

1205771 119864

0 1205772 1198601205772 119864

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119860

sdot 1205772 119864minus 1205771 119864

sdot 1205772 119860)

(35)

Similarly 119878Δ119863119874119864

is

119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119864

1205771 119863

0 1205772 1198641205772 119863

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119864

sdot 1205772 119863minus 1205771 119863

sdot 1205772 119864)

(36)






namely

119878∙

119860119862119861= 119878

10067040 119860119862119861minus 119878

Δ119860119874119861

11987810067040 119860119862119861

=12ang119860119874119861 sdot 119877

2=12ang119860119874119861 sdot 12


119878Δ119860119874119861

=12119860119861 sdot 119889 =


(37)


119888

(119894Δ119905)119892119903

(119894Δ119905)



Solution of

+

Begin

End

NoYes





Eq (20)

i lt N


i = i + 1

solving PosEi

Posf(0)

Pf(T) and Rs(T)

N

sumi=1

(PosEi)




Ntotal

(i)

(ii)(iii)

(iv)

(v)



No

No

No

Yes

Yes

Yes

Yes

No


N = TΔt

No

No

i = i + 1

j = 0

j = 0

i = i + 1

i = 1

i gt n



xi(NΔt)

xi isin Ωi


g(jΔt) gt 0

j lt N

j = j + 1



Ngenerated




Pos 119864119894 =

119860119894


total=119878119861119863119864

120587

=119878

10067040 119860119862119861119863minus 119878

Δ119860119874119864minus 119878

Δ119863119874119864minus 119878

∙

119860119862119861

120587

(38)








119894

119883119894(0) 119883

119894(Δ119905) 119883

119894(119873Δ119905)119873 = 119879Δ119905 in sequence


119894(119905) 119883119903

119894(119905)

and 120588119883119894



Ωi

(Xi minusXci )

119879

sdot(

11986211 11986212 1198621119873+1

11986221 11986222 1198622119873+1

119862119873+11 119862

119873+12 119862119873+1119873+1

)

minus1

(Xi minusXci )

le 1

119894 = 1 2 119899

(39)

where 119862119897119904= Cov

119883119894



119894(Δ119905)












d1

d2

F1(t)F2(t)

F3(t)l







119892 (119905) = 119892 (1198651 (119905) 1198652 (119905) 1198653 (119905) 119872criteria (119905))


sdot 1198653 (119905) 119905 isin [0 119879]

(40)










2(119905) 119865

3(119905)119872criteria(119905)

119865119888

1(119905) =

1

2sin(4120587119905 + 120587

4) +

5

2

119865119903

1(119905) = 119865

119888

1(119905) times 6120572

(unit N)

119865119888

2(119905) = minus cos(8120587119905 minus 120587

3) + 5

119865119903

2(119905) = 119865

119888

2(119905) times 5120572

(unit N)

119865119888

3(119905) = minus

3

4cos(6120587119905 + 2120587

5) +

3

2

119865119903

3(119905) = 119865

119888

3(119905) times 3120572

(unit N)

119872119888


5119905

119872119903

criteria (119905) =120572119905

2

100


1(119905) 119865

2(119905) 119865

3(119905)119872criteria(119905)

1205881198651

(1199051 119905

2) = 119890


(1199051 119905

2) = 119890



(1199051 119905

2) = 119890


(1199051 119905

2) =

1

2(1 + cos (6119896 sdot 1003816100381610038161003816t1 minus t2

1003816100381610038161003816))


119896120572 120572 = 10 120572 = 12 120572 = 14 120572 = 16 120572 = 18 120572 = 20119896 = 1 1 0986 09426 08931 08259 07595119896 = 2 1 09857 09424 08881 08155 07425119896 = 3 1 09854 09423 08821 08027 07249119896 = 4 1 09853 09419 08741 07915 07085119896 = 5 1 09852 09406 08696 07844 06972

10 2018161412070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

k = 1 k = 2k = 3

k = 4k = 5

120572 (T = 10 years)








1 2 3 4 5068

070

072

074

076

078

080

Tim

e-de

pend

ent r

eliab

ility





119904(119879) is widening










2(119905) 119865

3(119905) 119877


119865119888

1(119905) = 119865

119888

2(119905) = 2 sin(3120587119905 + 120587

3) + 4448

119865119903

1(119905) = 119865

119903

2(119905) = 119865

119888

1(119905) times 3120572

(unit KN)

119865119888

3(119905) = 7 cos (4120587119905 minus 120587

5) + 17792

119865119903

3(119905) = 119865

119888

3(119905) times 2120572

(unit KN)

119877119888

119879criteria(119905) = 375 minus15119905

8119877119903

119879criteria(119905) =120572119905

2

36

119877119888

119862criteria(119905) = minus100 +3119905

4119877119903

119862criteria(119905) =120572119905

2

32


1(119905) 119865

2(119905) 119865

3(119905) 119877


1205881198651

(1199051 119905

2) = 119890

minus(5|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198651(119905) and 119865

2(119905)120588

1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 120588


(1199051 119905

2) = 0

1

246

35

l l

X

Y

43

108

65

97

21

l

F1(t) F2(t)

F3(t)


119877119879criteria(119905) and 119877



119905 isin [0 119879]

(41)

1198922 (119905) = 119892 (119878119862max (119905) 119877119862criteria (119905))

=1003816100381610038161003816119877119862criteria (119905)

1003816100381610038161003816

minus1003816100381610038161003816119878119862max (1198651 (119905) 1198652 (119905) 1198653 (119905))

1003816100381610038161003816

119905 isin [0 119879]

(42)









12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 09096 0873 08329 07904 07369 0671412058811986511198652

= minus01 09085 08704 08319 07873 07327 0664712058811986511198652

= 0 0906 08684 08302 07842 07284 0663512058811986511198652

= 01 09053 08675 08296 07821 07272 065912058811986511198652

= 02 09047 08669 08276 07805 07238 06579


12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 1 09947 09581 08928 07899 0652112058811986511198652

= minus01 099941 09817 0929 08385 07075 0565812058811986511198652

= 0 09954 09603 08888 07762 0633 048212058811986511198652

= 01 09859 0935 08405 07057 05589 0402812058811986511198652

= 02 09706 09003 07858 06381 04884 03216


11986511198652is




(3) 12058811986511198652







15 2019181716065

070

075

080

085

090


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)


15 2019181716060

065

070

075

080

085

090

095

Tim

e-de

pend

ent r

eliab

ility




15 2019181716

04

05

06

07

08

09

10

Tim

e-de

pend

ent r

eliab

ility





119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










T

0

g(t)


g(t) = g(X(t))




] (119905)

= limΔ119905rarr 0

Pr (119892 (119905X (119905)) gt 0) cap (119892 (119905 + Δ119905X (119905 + Δ119905)) lt 0)Δ119905

(15)


] (119905)


Δ119905

(16)





119892 (0) 119892 (Δ119905) 119892 (2Δ119905) 119892 (119873Δ119905)

119873 =119879


(17)


119894

119892(119894Δ119905) gt 0 cap 119892((119894 + 1)Δ119905) lt 0


] (119894Δ119905) asympPos 119864

119894

Δ119905

=Pos (119892 (119894Δ119905) gt 0) cap (119892 ((119894 + 1) Δ119905) lt 0)

Δ119905

(18)


119894



=119860119894


total

(19)


119877119904(119879) = 1minus119875

119891(119879)


sum

119894=1(] (119894Δ119905) sdot Δ119905))

= 1minus(Pos (0) +119873

sum

119894=1(Pos 119864

119894))

= 1minus(Pos (0) +119873

sum

119894=1(119860


total))

(20)



(1199032radic2)119892119903

(119905) lt 119892119888

(119905) lt 119892119903


119894is

equivalent to

119864119894 119892 (119894Δ119905) gt 0cap119892 ((119894 + 1) Δ119905) lt 0 997904rArr 119892

119888

(119894Δ119905)

+ 119892119903

(119894Δ119905) sdot 1198801 gt 0cap119892119888

((119894 + 1) Δ119905)

+ 119892119903


(119894Δ119905)

119892119903 (119894Δ119905)

cap1198802 ltminus119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(21)



g((i + 1)Δt)

g((i + 1)Δt)

gc((i + 1)Δt)


gc(iΔt)

gr(iΔt)

g(iΔt) g(iΔt)

gr((i + 1)Δt)


gr((i + 1)Δt)

U2

r1

0

r2

U1

U1 = minusgc(iΔt)

gr(iΔt)

(a) (b)



119864119894997904rArr 1198801 gtminus

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap1198802 ltminus

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


(119894Δ119905)

119892119903 (119894Δ119905)


((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

997904rArr 1205772 lt119903111990321205771 +

radic21199032

sdot119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap 1205772 ltminus

119903111990321205771

minusradic21199032

sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(22)








(119894Δ119905)

119892119903 (119894Δ119905)= 0


(23)


(11990321 + 119903

22) sdot 120577


119892119888

(119894Δ119905)

119892119903 (119894Δ119905)sdot 1205772 + 2(

119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2

minus 11990321 = 0

(24)

1ζ

2ζ

O

A

D

C

B

E

R = 1d



1205772 119860=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)


119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(25)

1205772 119861=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)


119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(26)




obtained


1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0


(27)

we arrive at

(11990321 + 119903

22) sdot 120577

22 + 2radic21199032 sdot

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

sdot 1205772

+ 2(119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2minus 119903

21 = 0

(28)


1205772 119862=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(29)

1205772 119863=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(30)



determined




(119894Δ119905)

119892119903 (119894Δ119905)= 0

1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0(31)

Then we have1205771 119864

=minusradic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) + 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))

21199031

1205772 119864

=

radic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) minus 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))21199032

(32)





ang119860119874119863 = tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

) (33)

The area 11987810067040 119860119862119861119863

equals

11987810067040 119860119862119861119863

=12ang119860119874119863 sdot 119877

2=12ang119860119874119863 sdot 12

=12(tanminus1 (

10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

))

(34)



and 119878Δ119863119874119864



119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119860

1205771 119864

0 1205772 1198601205772 119864

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119860

sdot 1205772 119864minus 1205771 119864

sdot 1205772 119860)

(35)

Similarly 119878Δ119863119874119864

is

119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119864

1205771 119863

0 1205772 1198641205772 119863

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119864

sdot 1205772 119863minus 1205771 119863

sdot 1205772 119864)

(36)






namely

119878∙

119860119862119861= 119878

10067040 119860119862119861minus 119878

Δ119860119874119861

11987810067040 119860119862119861

=12ang119860119874119861 sdot 119877

2=12ang119860119874119861 sdot 12


119878Δ119860119874119861

=12119860119861 sdot 119889 =


(37)


119888

(119894Δ119905)119892119903

(119894Δ119905)



Solution of

+

Begin

End

NoYes





Eq (20)

i lt N


i = i + 1

solving PosEi

Posf(0)

Pf(T) and Rs(T)

N

sumi=1

(PosEi)




Ntotal

(i)

(ii)(iii)

(iv)

(v)



No

No

No

Yes

Yes

Yes

Yes

No


N = TΔt

No

No

i = i + 1

j = 0

j = 0

i = i + 1

i = 1

i gt n



xi(NΔt)

xi isin Ωi


g(jΔt) gt 0

j lt N

j = j + 1



Ngenerated




Pos 119864119894 =

119860119894


total=119878119861119863119864

120587

=119878

10067040 119860119862119861119863minus 119878

Δ119860119874119864minus 119878

Δ119863119874119864minus 119878

∙

119860119862119861

120587

(38)








119894

119883119894(0) 119883

119894(Δ119905) 119883

119894(119873Δ119905)119873 = 119879Δ119905 in sequence


119894(119905) 119883119903

119894(119905)

and 120588119883119894



Ωi

(Xi minusXci )

119879

sdot(

11986211 11986212 1198621119873+1

11986221 11986222 1198622119873+1

119862119873+11 119862

119873+12 119862119873+1119873+1

)

minus1

(Xi minusXci )

le 1

119894 = 1 2 119899

(39)

where 119862119897119904= Cov

119883119894



119894(Δ119905)












d1

d2

F1(t)F2(t)

F3(t)l







119892 (119905) = 119892 (1198651 (119905) 1198652 (119905) 1198653 (119905) 119872criteria (119905))


sdot 1198653 (119905) 119905 isin [0 119879]

(40)










2(119905) 119865

3(119905)119872criteria(119905)

119865119888

1(119905) =

1

2sin(4120587119905 + 120587

4) +

5

2

119865119903

1(119905) = 119865

119888

1(119905) times 6120572

(unit N)

119865119888

2(119905) = minus cos(8120587119905 minus 120587

3) + 5

119865119903

2(119905) = 119865

119888

2(119905) times 5120572

(unit N)

119865119888

3(119905) = minus

3

4cos(6120587119905 + 2120587

5) +

3

2

119865119903

3(119905) = 119865

119888

3(119905) times 3120572

(unit N)

119872119888


5119905

119872119903

criteria (119905) =120572119905

2

100


1(119905) 119865

2(119905) 119865

3(119905)119872criteria(119905)

1205881198651

(1199051 119905

2) = 119890


(1199051 119905

2) = 119890



(1199051 119905

2) = 119890


(1199051 119905

2) =

1

2(1 + cos (6119896 sdot 1003816100381610038161003816t1 minus t2

1003816100381610038161003816))


119896120572 120572 = 10 120572 = 12 120572 = 14 120572 = 16 120572 = 18 120572 = 20119896 = 1 1 0986 09426 08931 08259 07595119896 = 2 1 09857 09424 08881 08155 07425119896 = 3 1 09854 09423 08821 08027 07249119896 = 4 1 09853 09419 08741 07915 07085119896 = 5 1 09852 09406 08696 07844 06972

10 2018161412070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

k = 1 k = 2k = 3

k = 4k = 5

120572 (T = 10 years)








1 2 3 4 5068

070

072

074

076

078

080

Tim

e-de

pend

ent r

eliab

ility





119904(119879) is widening










2(119905) 119865

3(119905) 119877


119865119888

1(119905) = 119865

119888

2(119905) = 2 sin(3120587119905 + 120587

3) + 4448

119865119903

1(119905) = 119865

119903

2(119905) = 119865

119888

1(119905) times 3120572

(unit KN)

119865119888

3(119905) = 7 cos (4120587119905 minus 120587

5) + 17792

119865119903

3(119905) = 119865

119888

3(119905) times 2120572

(unit KN)

119877119888

119879criteria(119905) = 375 minus15119905

8119877119903

119879criteria(119905) =120572119905

2

36

119877119888

119862criteria(119905) = minus100 +3119905

4119877119903

119862criteria(119905) =120572119905

2

32


1(119905) 119865

2(119905) 119865

3(119905) 119877


1205881198651

(1199051 119905

2) = 119890

minus(5|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198651(119905) and 119865

2(119905)120588

1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 120588


(1199051 119905

2) = 0

1

246

35

l l

X

Y

43

108

65

97

21

l

F1(t) F2(t)

F3(t)


119877119879criteria(119905) and 119877



119905 isin [0 119879]

(41)

1198922 (119905) = 119892 (119878119862max (119905) 119877119862criteria (119905))

=1003816100381610038161003816119877119862criteria (119905)

1003816100381610038161003816

minus1003816100381610038161003816119878119862max (1198651 (119905) 1198652 (119905) 1198653 (119905))

1003816100381610038161003816

119905 isin [0 119879]

(42)









12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 09096 0873 08329 07904 07369 0671412058811986511198652

= minus01 09085 08704 08319 07873 07327 0664712058811986511198652

= 0 0906 08684 08302 07842 07284 0663512058811986511198652

= 01 09053 08675 08296 07821 07272 065912058811986511198652

= 02 09047 08669 08276 07805 07238 06579


12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 1 09947 09581 08928 07899 0652112058811986511198652

= minus01 099941 09817 0929 08385 07075 0565812058811986511198652

= 0 09954 09603 08888 07762 0633 048212058811986511198652

= 01 09859 0935 08405 07057 05589 0402812058811986511198652

= 02 09706 09003 07858 06381 04884 03216


11986511198652is




(3) 12058811986511198652







15 2019181716065

070

075

080

085

090


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)


15 2019181716060

065

070

075

080

085

090

095

Tim

e-de

pend

ent r

eliab

ility




15 2019181716

04

05

06

07

08

09

10

Tim

e-de

pend

ent r

eliab

ility





119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










g((i + 1)Δt)

g((i + 1)Δt)

gc((i + 1)Δt)


gc(iΔt)

gr(iΔt)

g(iΔt) g(iΔt)

gr((i + 1)Δt)


gr((i + 1)Δt)

U2

r1

0

r2

U1

U1 = minusgc(iΔt)

gr(iΔt)

(a) (b)



119864119894997904rArr 1198801 gtminus

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap1198802 ltminus

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


(119894Δ119905)

119892119903 (119894Δ119905)


((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

997904rArr 1205772 lt119903111990321205771 +

radic21199032

sdot119892119888

(119894Δ119905)

119892119903 (119894Δ119905)cap 1205772 ltminus

119903111990321205771

minusradic21199032

sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

(22)








(119894Δ119905)

119892119903 (119894Δ119905)= 0


(23)


(11990321 + 119903

22) sdot 120577


119892119888

(119894Δ119905)

119892119903 (119894Δ119905)sdot 1205772 + 2(

119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2

minus 11990321 = 0

(24)

1ζ

2ζ

O

A

D

C

B

E

R = 1d



1205772 119860=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)


119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(25)

1205772 119861=radic22(1199032 sdot

119892119888

(119894Δ119905)

119892119903 (119894Δ119905)


119892119888

(119894Δ119905)

119892119903 (119894Δ119905))

2+ 119903

21)

(26)




obtained


1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0


(27)

we arrive at

(11990321 + 119903

22) sdot 120577

22 + 2radic21199032 sdot

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

sdot 1205772

+ 2(119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2minus 119903

21 = 0

(28)


1205772 119862=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(29)

1205772 119863=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(30)



determined




(119894Δ119905)

119892119903 (119894Δ119905)= 0

1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0(31)

Then we have1205771 119864

=minusradic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) + 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))

21199031

1205772 119864

=

radic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) minus 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))21199032

(32)





ang119860119874119863 = tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

) (33)

The area 11987810067040 119860119862119861119863

equals

11987810067040 119860119862119861119863

=12ang119860119874119863 sdot 119877

2=12ang119860119874119863 sdot 12

=12(tanminus1 (

10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

))

(34)



and 119878Δ119863119874119864



119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119860

1205771 119864

0 1205772 1198601205772 119864

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119860

sdot 1205772 119864minus 1205771 119864

sdot 1205772 119860)

(35)

Similarly 119878Δ119863119874119864

is

119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119864

1205771 119863

0 1205772 1198641205772 119863

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119864

sdot 1205772 119863minus 1205771 119863

sdot 1205772 119864)

(36)






namely

119878∙

119860119862119861= 119878

10067040 119860119862119861minus 119878

Δ119860119874119861

11987810067040 119860119862119861

=12ang119860119874119861 sdot 119877

2=12ang119860119874119861 sdot 12


119878Δ119860119874119861

=12119860119861 sdot 119889 =


(37)


119888

(119894Δ119905)119892119903

(119894Δ119905)



Solution of

+

Begin

End

NoYes





Eq (20)

i lt N


i = i + 1

solving PosEi

Posf(0)

Pf(T) and Rs(T)

N

sumi=1

(PosEi)




Ntotal

(i)

(ii)(iii)

(iv)

(v)



No

No

No

Yes

Yes

Yes

Yes

No


N = TΔt

No

No

i = i + 1

j = 0

j = 0

i = i + 1

i = 1

i gt n



xi(NΔt)

xi isin Ωi


g(jΔt) gt 0

j lt N

j = j + 1



Ngenerated




Pos 119864119894 =

119860119894


total=119878119861119863119864

120587

=119878

10067040 119860119862119861119863minus 119878

Δ119860119874119864minus 119878

Δ119863119874119864minus 119878

∙

119860119862119861

120587

(38)








119894

119883119894(0) 119883

119894(Δ119905) 119883

119894(119873Δ119905)119873 = 119879Δ119905 in sequence


119894(119905) 119883119903

119894(119905)

and 120588119883119894



Ωi

(Xi minusXci )

119879

sdot(

11986211 11986212 1198621119873+1

11986221 11986222 1198622119873+1

119862119873+11 119862

119873+12 119862119873+1119873+1

)

minus1

(Xi minusXci )

le 1

119894 = 1 2 119899

(39)

where 119862119897119904= Cov

119883119894



119894(Δ119905)












d1

d2

F1(t)F2(t)

F3(t)l







119892 (119905) = 119892 (1198651 (119905) 1198652 (119905) 1198653 (119905) 119872criteria (119905))


sdot 1198653 (119905) 119905 isin [0 119879]

(40)










2(119905) 119865

3(119905)119872criteria(119905)

119865119888

1(119905) =

1

2sin(4120587119905 + 120587

4) +

5

2

119865119903

1(119905) = 119865

119888

1(119905) times 6120572

(unit N)

119865119888

2(119905) = minus cos(8120587119905 minus 120587

3) + 5

119865119903

2(119905) = 119865

119888

2(119905) times 5120572

(unit N)

119865119888

3(119905) = minus

3

4cos(6120587119905 + 2120587

5) +

3

2

119865119903

3(119905) = 119865

119888

3(119905) times 3120572

(unit N)

119872119888


5119905

119872119903

criteria (119905) =120572119905

2

100


1(119905) 119865

2(119905) 119865

3(119905)119872criteria(119905)

1205881198651

(1199051 119905

2) = 119890


(1199051 119905

2) = 119890



(1199051 119905

2) = 119890


(1199051 119905

2) =

1

2(1 + cos (6119896 sdot 1003816100381610038161003816t1 minus t2

1003816100381610038161003816))


119896120572 120572 = 10 120572 = 12 120572 = 14 120572 = 16 120572 = 18 120572 = 20119896 = 1 1 0986 09426 08931 08259 07595119896 = 2 1 09857 09424 08881 08155 07425119896 = 3 1 09854 09423 08821 08027 07249119896 = 4 1 09853 09419 08741 07915 07085119896 = 5 1 09852 09406 08696 07844 06972

10 2018161412070

075

080

085

090

095

100


Tim

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pend

ent r

eliab

ility

k = 1 k = 2k = 3

k = 4k = 5

120572 (T = 10 years)








1 2 3 4 5068

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072

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078

080

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ility





119904(119879) is widening










2(119905) 119865

3(119905) 119877


119865119888

1(119905) = 119865

119888

2(119905) = 2 sin(3120587119905 + 120587

3) + 4448

119865119903

1(119905) = 119865

119903

2(119905) = 119865

119888

1(119905) times 3120572

(unit KN)

119865119888

3(119905) = 7 cos (4120587119905 minus 120587

5) + 17792

119865119903

3(119905) = 119865

119888

3(119905) times 2120572

(unit KN)

119877119888

119879criteria(119905) = 375 minus15119905

8119877119903

119879criteria(119905) =120572119905

2

36

119877119888

119862criteria(119905) = minus100 +3119905

4119877119903

119862criteria(119905) =120572119905

2

32


1(119905) 119865

2(119905) 119865

3(119905) 119877


1205881198651

(1199051 119905

2) = 119890

minus(5|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198651(119905) and 119865

2(119905)120588

1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 120588


(1199051 119905

2) = 0

1

246

35

l l

X

Y

43

108

65

97

21

l

F1(t) F2(t)

F3(t)


119877119879criteria(119905) and 119877



119905 isin [0 119879]

(41)

1198922 (119905) = 119892 (119878119862max (119905) 119877119862criteria (119905))

=1003816100381610038161003816119877119862criteria (119905)

1003816100381610038161003816

minus1003816100381610038161003816119878119862max (1198651 (119905) 1198652 (119905) 1198653 (119905))

1003816100381610038161003816

119905 isin [0 119879]

(42)









12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 09096 0873 08329 07904 07369 0671412058811986511198652

= minus01 09085 08704 08319 07873 07327 0664712058811986511198652

= 0 0906 08684 08302 07842 07284 0663512058811986511198652

= 01 09053 08675 08296 07821 07272 065912058811986511198652

= 02 09047 08669 08276 07805 07238 06579


12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 1 09947 09581 08928 07899 0652112058811986511198652

= minus01 099941 09817 0929 08385 07075 0565812058811986511198652

= 0 09954 09603 08888 07762 0633 048212058811986511198652

= 01 09859 0935 08405 07057 05589 0402812058811986511198652

= 02 09706 09003 07858 06381 04884 03216


11986511198652is




(3) 12058811986511198652







15 2019181716065

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eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)


15 2019181716060

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095

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15 2019181716

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05

06

07

08

09

10

Tim

e-de

pend

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eliab

ility





119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












obtained


1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0


(27)

we arrive at

(11990321 + 119903

22) sdot 120577

22 + 2radic21199032 sdot

119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

sdot 1205772

+ 2(119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2minus 119903

21 = 0

(28)


1205772 119862=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(29)

1205772 119863=minusradic22

(1199032 sdot119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)


119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

)

2+ 119903

21)

(30)



determined




(119894Δ119905)

119892119903 (119894Δ119905)= 0

1198972 11990321205772 + 11990311205771 +radic2119892119888

((119894 + 1) Δ119905)119892119903 ((119894 + 1) Δ119905)

= 0(31)

Then we have1205771 119864

=minusradic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) + 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))

21199031

1205772 119864

=

radic2 (119892119888 (119894Δ119905) 119892119903 (119894Δ119905) minus 119892119888 ((119894 + 1) Δ119905) 119892119903 ((119894 + 1) Δ119905))21199032

(32)





ang119860119874119863 = tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

) (33)

The area 11987810067040 119860119862119861119863

equals

11987810067040 119860119862119861119863

=12ang119860119874119863 sdot 119877

2=12ang119860119874119863 sdot 12

=12(tanminus1 (

10038161003816100381610038161003816100381610038161003816

1205772 119860

1205771 119860

10038161003816100381610038161003816100381610038161003816

) + tanminus1 (10038161003816100381610038161003816100381610038161003816

1205772 119863

1205771 119863

10038161003816100381610038161003816100381610038161003816

))

(34)



and 119878Δ119863119874119864



119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119860

1205771 119864

0 1205772 1198601205772 119864

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119860

sdot 1205772 119864minus 1205771 119864

sdot 1205772 119860)

(35)

Similarly 119878Δ119863119874119864

is

119878Δ119860119874119864

=12

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1 10 1205771 119864

1205771 119863

0 1205772 1198641205772 119863

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=12(1205771 119864

sdot 1205772 119863minus 1205771 119863

sdot 1205772 119864)

(36)






namely

119878∙

119860119862119861= 119878

10067040 119860119862119861minus 119878

Δ119860119874119861

11987810067040 119860119862119861

=12ang119860119874119861 sdot 119877

2=12ang119860119874119861 sdot 12


119878Δ119860119874119861

=12119860119861 sdot 119889 =


(37)


119888

(119894Δ119905)119892119903

(119894Δ119905)



Solution of

+

Begin

End

NoYes





Eq (20)

i lt N


i = i + 1

solving PosEi

Posf(0)

Pf(T) and Rs(T)

N

sumi=1

(PosEi)




Ntotal

(i)

(ii)(iii)

(iv)

(v)



No

No

No

Yes

Yes

Yes

Yes

No


N = TΔt

No

No

i = i + 1

j = 0

j = 0

i = i + 1

i = 1

i gt n



xi(NΔt)

xi isin Ωi


g(jΔt) gt 0

j lt N

j = j + 1



Ngenerated




Pos 119864119894 =

119860119894


total=119878119861119863119864

120587

=119878

10067040 119860119862119861119863minus 119878

Δ119860119874119864minus 119878

Δ119863119874119864minus 119878

∙

119860119862119861

120587

(38)








119894

119883119894(0) 119883

119894(Δ119905) 119883

119894(119873Δ119905)119873 = 119879Δ119905 in sequence


119894(119905) 119883119903

119894(119905)

and 120588119883119894



Ωi

(Xi minusXci )

119879

sdot(

11986211 11986212 1198621119873+1

11986221 11986222 1198622119873+1

119862119873+11 119862

119873+12 119862119873+1119873+1

)

minus1

(Xi minusXci )

le 1

119894 = 1 2 119899

(39)

where 119862119897119904= Cov

119883119894



119894(Δ119905)












d1

d2

F1(t)F2(t)

F3(t)l







119892 (119905) = 119892 (1198651 (119905) 1198652 (119905) 1198653 (119905) 119872criteria (119905))


sdot 1198653 (119905) 119905 isin [0 119879]

(40)










2(119905) 119865

3(119905)119872criteria(119905)

119865119888

1(119905) =

1

2sin(4120587119905 + 120587

4) +

5

2

119865119903

1(119905) = 119865

119888

1(119905) times 6120572

(unit N)

119865119888

2(119905) = minus cos(8120587119905 minus 120587

3) + 5

119865119903

2(119905) = 119865

119888

2(119905) times 5120572

(unit N)

119865119888

3(119905) = minus

3

4cos(6120587119905 + 2120587

5) +

3

2

119865119903

3(119905) = 119865

119888

3(119905) times 3120572

(unit N)

119872119888


5119905

119872119903

criteria (119905) =120572119905

2

100


1(119905) 119865

2(119905) 119865

3(119905)119872criteria(119905)

1205881198651

(1199051 119905

2) = 119890


(1199051 119905

2) = 119890



(1199051 119905

2) = 119890


(1199051 119905

2) =

1

2(1 + cos (6119896 sdot 1003816100381610038161003816t1 minus t2

1003816100381610038161003816))


119896120572 120572 = 10 120572 = 12 120572 = 14 120572 = 16 120572 = 18 120572 = 20119896 = 1 1 0986 09426 08931 08259 07595119896 = 2 1 09857 09424 08881 08155 07425119896 = 3 1 09854 09423 08821 08027 07249119896 = 4 1 09853 09419 08741 07915 07085119896 = 5 1 09852 09406 08696 07844 06972

10 2018161412070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

k = 1 k = 2k = 3

k = 4k = 5

120572 (T = 10 years)








1 2 3 4 5068

070

072

074

076

078

080

Tim

e-de

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eliab

ility





119904(119879) is widening










2(119905) 119865

3(119905) 119877


119865119888

1(119905) = 119865

119888

2(119905) = 2 sin(3120587119905 + 120587

3) + 4448

119865119903

1(119905) = 119865

119903

2(119905) = 119865

119888

1(119905) times 3120572

(unit KN)

119865119888

3(119905) = 7 cos (4120587119905 minus 120587

5) + 17792

119865119903

3(119905) = 119865

119888

3(119905) times 2120572

(unit KN)

119877119888

119879criteria(119905) = 375 minus15119905

8119877119903

119879criteria(119905) =120572119905

2

36

119877119888

119862criteria(119905) = minus100 +3119905

4119877119903

119862criteria(119905) =120572119905

2

32


1(119905) 119865

2(119905) 119865

3(119905) 119877


1205881198651

(1199051 119905

2) = 119890

minus(5|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198651(119905) and 119865

2(119905)120588

1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 120588


(1199051 119905

2) = 0

1

246

35

l l

X

Y

43

108

65

97

21

l

F1(t) F2(t)

F3(t)


119877119879criteria(119905) and 119877



119905 isin [0 119879]

(41)

1198922 (119905) = 119892 (119878119862max (119905) 119877119862criteria (119905))

=1003816100381610038161003816119877119862criteria (119905)

1003816100381610038161003816

minus1003816100381610038161003816119878119862max (1198651 (119905) 1198652 (119905) 1198653 (119905))

1003816100381610038161003816

119905 isin [0 119879]

(42)









12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 09096 0873 08329 07904 07369 0671412058811986511198652

= minus01 09085 08704 08319 07873 07327 0664712058811986511198652

= 0 0906 08684 08302 07842 07284 0663512058811986511198652

= 01 09053 08675 08296 07821 07272 065912058811986511198652

= 02 09047 08669 08276 07805 07238 06579


12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 1 09947 09581 08928 07899 0652112058811986511198652

= minus01 099941 09817 0929 08385 07075 0565812058811986511198652

= 0 09954 09603 08888 07762 0633 048212058811986511198652

= 01 09859 0935 08405 07057 05589 0402812058811986511198652

= 02 09706 09003 07858 06381 04884 03216


11986511198652is




(3) 12058811986511198652







15 2019181716065

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075

080

085

090


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)


15 2019181716060

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095

Tim

e-de

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eliab

ility




15 2019181716

04

05

06

07

08

09

10

Tim

e-de

pend

ent r

eliab

ility





119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Solution of

+

Begin

End

NoYes





Eq (20)

i lt N


i = i + 1

solving PosEi

Posf(0)

Pf(T) and Rs(T)

N

sumi=1

(PosEi)




Ntotal

(i)

(ii)(iii)

(iv)

(v)



No

No

No

Yes

Yes

Yes

Yes

No


N = TΔt

No

No

i = i + 1

j = 0

j = 0

i = i + 1

i = 1

i gt n



xi(NΔt)

xi isin Ωi


g(jΔt) gt 0

j lt N

j = j + 1



Ngenerated




Pos 119864119894 =

119860119894


total=119878119861119863119864

120587

=119878

10067040 119860119862119861119863minus 119878

Δ119860119874119864minus 119878

Δ119863119874119864minus 119878

∙

119860119862119861

120587

(38)








119894

119883119894(0) 119883

119894(Δ119905) 119883

119894(119873Δ119905)119873 = 119879Δ119905 in sequence


119894(119905) 119883119903

119894(119905)

and 120588119883119894



Ωi

(Xi minusXci )

119879

sdot(

11986211 11986212 1198621119873+1

11986221 11986222 1198622119873+1

119862119873+11 119862

119873+12 119862119873+1119873+1

)

minus1

(Xi minusXci )

le 1

119894 = 1 2 119899

(39)

where 119862119897119904= Cov

119883119894



119894(Δ119905)












d1

d2

F1(t)F2(t)

F3(t)l







119892 (119905) = 119892 (1198651 (119905) 1198652 (119905) 1198653 (119905) 119872criteria (119905))


sdot 1198653 (119905) 119905 isin [0 119879]

(40)










2(119905) 119865

3(119905)119872criteria(119905)

119865119888

1(119905) =

1

2sin(4120587119905 + 120587

4) +

5

2

119865119903

1(119905) = 119865

119888

1(119905) times 6120572

(unit N)

119865119888

2(119905) = minus cos(8120587119905 minus 120587

3) + 5

119865119903

2(119905) = 119865

119888

2(119905) times 5120572

(unit N)

119865119888

3(119905) = minus

3

4cos(6120587119905 + 2120587

5) +

3

2

119865119903

3(119905) = 119865

119888

3(119905) times 3120572

(unit N)

119872119888


5119905

119872119903

criteria (119905) =120572119905

2

100


1(119905) 119865

2(119905) 119865

3(119905)119872criteria(119905)

1205881198651

(1199051 119905

2) = 119890


(1199051 119905

2) = 119890



(1199051 119905

2) = 119890


(1199051 119905

2) =

1

2(1 + cos (6119896 sdot 1003816100381610038161003816t1 minus t2

1003816100381610038161003816))


119896120572 120572 = 10 120572 = 12 120572 = 14 120572 = 16 120572 = 18 120572 = 20119896 = 1 1 0986 09426 08931 08259 07595119896 = 2 1 09857 09424 08881 08155 07425119896 = 3 1 09854 09423 08821 08027 07249119896 = 4 1 09853 09419 08741 07915 07085119896 = 5 1 09852 09406 08696 07844 06972

10 2018161412070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

k = 1 k = 2k = 3

k = 4k = 5

120572 (T = 10 years)








1 2 3 4 5068

070

072

074

076

078

080

Tim

e-de

pend

ent r

eliab

ility





119904(119879) is widening










2(119905) 119865

3(119905) 119877


119865119888

1(119905) = 119865

119888

2(119905) = 2 sin(3120587119905 + 120587

3) + 4448

119865119903

1(119905) = 119865

119903

2(119905) = 119865

119888

1(119905) times 3120572

(unit KN)

119865119888

3(119905) = 7 cos (4120587119905 minus 120587

5) + 17792

119865119903

3(119905) = 119865

119888

3(119905) times 2120572

(unit KN)

119877119888

119879criteria(119905) = 375 minus15119905

8119877119903

119879criteria(119905) =120572119905

2

36

119877119888

119862criteria(119905) = minus100 +3119905

4119877119903

119862criteria(119905) =120572119905

2

32


1(119905) 119865

2(119905) 119865

3(119905) 119877


1205881198651

(1199051 119905

2) = 119890

minus(5|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198651(119905) and 119865

2(119905)120588

1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 120588


(1199051 119905

2) = 0

1

246

35

l l

X

Y

43

108

65

97

21

l

F1(t) F2(t)

F3(t)


119877119879criteria(119905) and 119877



119905 isin [0 119879]

(41)

1198922 (119905) = 119892 (119878119862max (119905) 119877119862criteria (119905))

=1003816100381610038161003816119877119862criteria (119905)

1003816100381610038161003816

minus1003816100381610038161003816119878119862max (1198651 (119905) 1198652 (119905) 1198653 (119905))

1003816100381610038161003816

119905 isin [0 119879]

(42)









12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 09096 0873 08329 07904 07369 0671412058811986511198652

= minus01 09085 08704 08319 07873 07327 0664712058811986511198652

= 0 0906 08684 08302 07842 07284 0663512058811986511198652

= 01 09053 08675 08296 07821 07272 065912058811986511198652

= 02 09047 08669 08276 07805 07238 06579


12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 1 09947 09581 08928 07899 0652112058811986511198652

= minus01 099941 09817 0929 08385 07075 0565812058811986511198652

= 0 09954 09603 08888 07762 0633 048212058811986511198652

= 01 09859 0935 08405 07057 05589 0402812058811986511198652

= 02 09706 09003 07858 06381 04884 03216


11986511198652is




(3) 12058811986511198652







15 2019181716065

070

075

080

085

090


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)


15 2019181716060

065

070

075

080

085

090

095

Tim

e-de

pend

ent r

eliab

ility




15 2019181716

04

05

06

07

08

09

10

Tim

e-de

pend

ent r

eliab

ility





119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Ωi

(Xi minusXci )

119879

sdot(

11986211 11986212 1198621119873+1

11986221 11986222 1198622119873+1

119862119873+11 119862

119873+12 119862119873+1119873+1

)

minus1

(Xi minusXci )

le 1

119894 = 1 2 119899

(39)

where 119862119897119904= Cov

119883119894



119894(Δ119905)












d1

d2

F1(t)F2(t)

F3(t)l







119892 (119905) = 119892 (1198651 (119905) 1198652 (119905) 1198653 (119905) 119872criteria (119905))


sdot 1198653 (119905) 119905 isin [0 119879]

(40)










2(119905) 119865

3(119905)119872criteria(119905)

119865119888

1(119905) =

1

2sin(4120587119905 + 120587

4) +

5

2

119865119903

1(119905) = 119865

119888

1(119905) times 6120572

(unit N)

119865119888

2(119905) = minus cos(8120587119905 minus 120587

3) + 5

119865119903

2(119905) = 119865

119888

2(119905) times 5120572

(unit N)

119865119888

3(119905) = minus

3

4cos(6120587119905 + 2120587

5) +

3

2

119865119903

3(119905) = 119865

119888

3(119905) times 3120572

(unit N)

119872119888


5119905

119872119903

criteria (119905) =120572119905

2

100


1(119905) 119865

2(119905) 119865

3(119905)119872criteria(119905)

1205881198651

(1199051 119905

2) = 119890


(1199051 119905

2) = 119890



(1199051 119905

2) = 119890


(1199051 119905

2) =

1

2(1 + cos (6119896 sdot 1003816100381610038161003816t1 minus t2

1003816100381610038161003816))


119896120572 120572 = 10 120572 = 12 120572 = 14 120572 = 16 120572 = 18 120572 = 20119896 = 1 1 0986 09426 08931 08259 07595119896 = 2 1 09857 09424 08881 08155 07425119896 = 3 1 09854 09423 08821 08027 07249119896 = 4 1 09853 09419 08741 07915 07085119896 = 5 1 09852 09406 08696 07844 06972

10 2018161412070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

k = 1 k = 2k = 3

k = 4k = 5

120572 (T = 10 years)








1 2 3 4 5068

070

072

074

076

078

080

Tim

e-de

pend

ent r

eliab

ility





119904(119879) is widening










2(119905) 119865

3(119905) 119877


119865119888

1(119905) = 119865

119888

2(119905) = 2 sin(3120587119905 + 120587

3) + 4448

119865119903

1(119905) = 119865

119903

2(119905) = 119865

119888

1(119905) times 3120572

(unit KN)

119865119888

3(119905) = 7 cos (4120587119905 minus 120587

5) + 17792

119865119903

3(119905) = 119865

119888

3(119905) times 2120572

(unit KN)

119877119888

119879criteria(119905) = 375 minus15119905

8119877119903

119879criteria(119905) =120572119905

2

36

119877119888

119862criteria(119905) = minus100 +3119905

4119877119903

119862criteria(119905) =120572119905

2

32


1(119905) 119865

2(119905) 119865

3(119905) 119877


1205881198651

(1199051 119905

2) = 119890

minus(5|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198651(119905) and 119865

2(119905)120588

1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 120588


(1199051 119905

2) = 0

1

246

35

l l

X

Y

43

108

65

97

21

l

F1(t) F2(t)

F3(t)


119877119879criteria(119905) and 119877



119905 isin [0 119879]

(41)

1198922 (119905) = 119892 (119878119862max (119905) 119877119862criteria (119905))

=1003816100381610038161003816119877119862criteria (119905)

1003816100381610038161003816

minus1003816100381610038161003816119878119862max (1198651 (119905) 1198652 (119905) 1198653 (119905))

1003816100381610038161003816

119905 isin [0 119879]

(42)









12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 09096 0873 08329 07904 07369 0671412058811986511198652

= minus01 09085 08704 08319 07873 07327 0664712058811986511198652

= 0 0906 08684 08302 07842 07284 0663512058811986511198652

= 01 09053 08675 08296 07821 07272 065912058811986511198652

= 02 09047 08669 08276 07805 07238 06579


12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 1 09947 09581 08928 07899 0652112058811986511198652

= minus01 099941 09817 0929 08385 07075 0565812058811986511198652

= 0 09954 09603 08888 07762 0633 048212058811986511198652

= 01 09859 0935 08405 07057 05589 0402812058811986511198652

= 02 09706 09003 07858 06381 04884 03216


11986511198652is




(3) 12058811986511198652







15 2019181716065

070

075

080

085

090


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)


15 2019181716060

065

070

075

080

085

090

095

Tim

e-de

pend

ent r

eliab

ility




15 2019181716

04

05

06

07

08

09

10

Tim

e-de

pend

ent r

eliab

ility





119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2(119905) 119865

3(119905)119872criteria(119905)

119865119888

1(119905) =

1

2sin(4120587119905 + 120587

4) +

5

2

119865119903

1(119905) = 119865

119888

1(119905) times 6120572

(unit N)

119865119888

2(119905) = minus cos(8120587119905 minus 120587

3) + 5

119865119903

2(119905) = 119865

119888

2(119905) times 5120572

(unit N)

119865119888

3(119905) = minus

3

4cos(6120587119905 + 2120587

5) +

3

2

119865119903

3(119905) = 119865

119888

3(119905) times 3120572

(unit N)

119872119888


5119905

119872119903

criteria (119905) =120572119905

2

100


1(119905) 119865

2(119905) 119865

3(119905)119872criteria(119905)

1205881198651

(1199051 119905

2) = 119890


(1199051 119905

2) = 119890



(1199051 119905

2) = 119890


(1199051 119905

2) =

1

2(1 + cos (6119896 sdot 1003816100381610038161003816t1 minus t2

1003816100381610038161003816))


119896120572 120572 = 10 120572 = 12 120572 = 14 120572 = 16 120572 = 18 120572 = 20119896 = 1 1 0986 09426 08931 08259 07595119896 = 2 1 09857 09424 08881 08155 07425119896 = 3 1 09854 09423 08821 08027 07249119896 = 4 1 09853 09419 08741 07915 07085119896 = 5 1 09852 09406 08696 07844 06972

10 2018161412070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

k = 1 k = 2k = 3

k = 4k = 5

120572 (T = 10 years)








1 2 3 4 5068

070

072

074

076

078

080

Tim

e-de

pend

ent r

eliab

ility





119904(119879) is widening










2(119905) 119865

3(119905) 119877


119865119888

1(119905) = 119865

119888

2(119905) = 2 sin(3120587119905 + 120587

3) + 4448

119865119903

1(119905) = 119865

119903

2(119905) = 119865

119888

1(119905) times 3120572

(unit KN)

119865119888

3(119905) = 7 cos (4120587119905 minus 120587

5) + 17792

119865119903

3(119905) = 119865

119888

3(119905) times 2120572

(unit KN)

119877119888

119879criteria(119905) = 375 minus15119905

8119877119903

119879criteria(119905) =120572119905

2

36

119877119888

119862criteria(119905) = minus100 +3119905

4119877119903

119862criteria(119905) =120572119905

2

32


1(119905) 119865

2(119905) 119865

3(119905) 119877


1205881198651

(1199051 119905

2) = 119890

minus(5|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198651(119905) and 119865

2(119905)120588

1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 120588


(1199051 119905

2) = 0

1

246

35

l l

X

Y

43

108

65

97

21

l

F1(t) F2(t)

F3(t)


119877119879criteria(119905) and 119877



119905 isin [0 119879]

(41)

1198922 (119905) = 119892 (119878119862max (119905) 119877119862criteria (119905))

=1003816100381610038161003816119877119862criteria (119905)

1003816100381610038161003816

minus1003816100381610038161003816119878119862max (1198651 (119905) 1198652 (119905) 1198653 (119905))

1003816100381610038161003816

119905 isin [0 119879]

(42)









12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 09096 0873 08329 07904 07369 0671412058811986511198652

= minus01 09085 08704 08319 07873 07327 0664712058811986511198652

= 0 0906 08684 08302 07842 07284 0663512058811986511198652

= 01 09053 08675 08296 07821 07272 065912058811986511198652

= 02 09047 08669 08276 07805 07238 06579


12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 1 09947 09581 08928 07899 0652112058811986511198652

= minus01 099941 09817 0929 08385 07075 0565812058811986511198652

= 0 09954 09603 08888 07762 0633 048212058811986511198652

= 01 09859 0935 08405 07057 05589 0402812058811986511198652

= 02 09706 09003 07858 06381 04884 03216


11986511198652is




(3) 12058811986511198652







15 2019181716065

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ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)


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085

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095

Tim

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eliab

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15 2019181716

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05

06

07

08

09

10

Tim

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119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2(119905) 119865

3(119905) 119877


119865119888

1(119905) = 119865

119888

2(119905) = 2 sin(3120587119905 + 120587

3) + 4448

119865119903

1(119905) = 119865

119903

2(119905) = 119865

119888

1(119905) times 3120572

(unit KN)

119865119888

3(119905) = 7 cos (4120587119905 minus 120587

5) + 17792

119865119903

3(119905) = 119865

119888

3(119905) times 2120572

(unit KN)

119877119888

119879criteria(119905) = 375 minus15119905

8119877119903

119879criteria(119905) =120572119905

2

36

119877119888

119862criteria(119905) = minus100 +3119905

4119877119903

119862criteria(119905) =120572119905

2

32


1(119905) 119865

2(119905) 119865

3(119905) 119877


1205881198651

(1199051 119905

2) = 119890

minus(5|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198651(119905) and 119865

2(119905)120588

1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 120588


(1199051 119905

2) = 0

1

246

35

l l

X

Y

43

108

65

97

21

l

F1(t) F2(t)

F3(t)


119877119879criteria(119905) and 119877



119905 isin [0 119879]

(41)

1198922 (119905) = 119892 (119878119862max (119905) 119877119862criteria (119905))

=1003816100381610038161003816119877119862criteria (119905)

1003816100381610038161003816

minus1003816100381610038161003816119878119862max (1198651 (119905) 1198652 (119905) 1198653 (119905))

1003816100381610038161003816

119905 isin [0 119879]

(42)









12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 09096 0873 08329 07904 07369 0671412058811986511198652

= minus01 09085 08704 08319 07873 07327 0664712058811986511198652

= 0 0906 08684 08302 07842 07284 0663512058811986511198652

= 01 09053 08675 08296 07821 07272 065912058811986511198652

= 02 09047 08669 08276 07805 07238 06579


12058811986511198652

120572 120572 = 15 120572 = 16 120572 = 17 120572 = 18 120572 = 19 120572 = 20

12058811986511198652

= minus02 1 09947 09581 08928 07899 0652112058811986511198652

= minus01 099941 09817 0929 08385 07075 0565812058811986511198652

= 0 09954 09603 08888 07762 0633 048212058811986511198652

= 01 09859 0935 08405 07057 05589 0402812058811986511198652

= 02 09706 09003 07858 06381 04884 03216


11986511198652is




(3) 12058811986511198652







15 2019181716065

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120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)


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095

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eliab

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15 2019181716

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06

07

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10

Tim

e-de

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eliab

ility





119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










15 2019181716065

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075

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085

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Tim

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eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02


(member 4)

15 2019181716


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)

120588F1F2 = minus02120588F1F2 = minus01

120588F1F2 = 0

120588F1F2 = 01

120588F1F2 = 02

03

04

05

06

07

08

09

10


(member 8)


15 2019181716060

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070

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080

085

090

095

Tim

e-de

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eliab

ility




15 2019181716

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05

06

07

08

09

10

Tim

e-de

pend

ent r

eliab

ility





119892 (119905) = 119892 (120590max (119905) 120590criteria (119905))


119905 isin [0 119879]

(43)


X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










X

Z

Y

Z

Y

Clamped supported



Shear force F3(t)








119904(119879) = 1)


119904(119879) = 07538 under the

01 100908070605040302070

075

080

085

090

095

100


Tim

e-de

pend

ent r

eliab

ility

120572 (T = 10 years)






2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2(119905) 119865

3(119905) 120590criteria(119905)

119865119888

1(119905) = 15 sin(4120587119905 + 3120587

4) + 60 119865

119888

2(119905) = 50 cos(3120587119905 + 120587

2) + 150 119865

119888

3(119905) = 10 sin(2120587119905 minus 3120587

5) + 50 (Unit KN)

119865119903

1(119905) = 119865

119888

1(119905) times 3 119865

119903

2(119905) = 119865

119888

2(119905) times 4 119865

119903

3(119905) = 119865

119888

3(119905) times 50

120590119888

criteria(119905) = 110119890minus(120572119905)

+ 290 120590119903



2(119905) 119865

3(119905) 120590criteria(119905)

1205881198651

(1199051 119905

2) = 119890

minus(3|t1minust2 |) 1205881198652

(1199051 119905

2) = 119890


1198653

(1199051 119905

2) = 119890


(1199051 119905

2) = 05

12058811986511198652

(1199051 119905

2) =

1

4(2 + cos (2 10038161003816100381610038161199051 minus 1199052

1003816100381610038161003816)) 12058811986511198653

(1199051 119905

2) =

1

6(4119890


(1199051 119905

2) =

8 minus 119905

20


120572 120572 = 01 120572 = 02 120572 = 03 120572 = 04 120572 = 05 120572 = 06 120572 = 07 120572 = 08 120572 = 09 120572 = 10

119896 = 1 1 1 09387 088 08211 07869 0769 07602 07556 07538






7 Conclusions








Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Time-Dependent Reliability Modeling and...

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