YU Ying-xia,LIU Feng-jun,ZHANG Wei,LIANG Bin

(School of Planning and Architectural Engineering,Henan University of Science and Technology,Luoyang 471003,China)

1 Introduction

First order reliability method(FORM)[1,2]is often applied to analyze the reliability of structure.The fundamental principle is that the function is derived using first order Taylor series expansion at the design point in the independent standard normal space and the failure state function is replaced by linear function to obtain an approximate solutions.There are two problems in using FORM.Firstly,the failure state function should be expressed in definite mathematic expressions.But it is difficult to give the explicit expressions in practical projects because the nonlinearity between input and output variables.Secondly,a quite great calculation error of reliability would happen if the nonlinear degree of failure state function is higher at the design point.The response surface method(RSM)[3-5]can solve above problems very well.The second problem can been solved effectively by second order reliability method(SORM)[1,2],but SORM should been used based on strict calculation and tends to be praise highly by the theory of scholar.

In this paper,the strength of ring-stiffened cylindrical shell is analyzed by using the finite element method and the failure state function is established based on the RSM,so the first above problem is solved.By studying the structural reliability theory,the FORM is combined with RSM(RSM-FORM)and improved.By all random variables being converted into independent and normalized parameters,the quadratic term of the function is kept and the computational accuracy of RSM-FORM is approximate with SORM.So the above problems are solved.The reliability of ring-stiffened cylindrical shell is analyzed with RSM-FORM and the result indicates that RSM-FORM has a better calculation precision.

2 Theoretical study of RSM-FORM

In general,because the stresses of a cylinder structure are difficult to be impressed by explicit expression,an implicit expression is used while the reliability index of ring-stiffened cylindrical shell being calculated.The RSM,as a method developed in these years,is an effective method to fit the failure state function.Through a series of finite element numerical calculation,a response surface equation can be established to replace the real unknown failure state function based on RSM.The response surface equation is often presented with a quadratic polynomial excluding cross terms that is shown in Eq.(1)and there are 2n+1 undetermined coefficients.

where zi,n, α0, αiand βiare,respectively,the ist random variable,the number of random variables and the undetermined coefficients that are determined with finite element analysis.

2.1 Correlations among random variables

2.2 Normalizing and independent of random variables

yiis normalized on the condition of normal equivalent and the mean valuesand variancesof equivalent normal variables can be calculated in Eq.(3)and Eq.(4).

The correlations among random variables are not changed because of normal transformation and the foregoing random variables are still uncorrelated[1].For normal random variable,non-relevance and independent is equivalent[1],so the random variables are independent of each other in Eq.(2).Through the above transformation,the random variables are all standard normal variables independent of each other.So FORM meets the applicable conditions.

2.3 Calculating of reliability index

Based on the above theory,Eq.(5)can be used to calculate the reliability indexes and Eq.(6)and Eq.(7)can be used to calculate the coordinate of design point （p*）.

3 Application of RSM-FORM

There are a lot of factors influencing the strength of ring-stiffened cylindrical shell,including material performance,structure size,and so on[6-7].By sensitivity analysis,rib spacing（L）,shell thickness（T）,inner diameter（R）,even outside pressure（P）and material yield strength （σs）are seen as random variables.The mean values of random variables followed by 600 mm,24 mm,3200 mm,3.92 MPa and 649.25 MPa.Based on structure strength analysis,27 sets of valid data[6]are selected and the failure state function(Eq.(8)is established by using the least square method.

The multiple correlation coefficient[8]is 0.99999 in Eq.(8)and the failure state function fits the data of the example,so it could be accepted.

Substituting into Eq.(8)yields as Eq.(9).

The first iteration process is as follows:

(1)Assuming the initial iteration point

(2)Normalizing the parameters

If yi=,then yiis transformed to normal distribution using Eqs.(3)and(4).

(3)Calculating the reliability index

(4)Calculating the coordinate of new design calculation point

Then the new design calculation point is used to calculate the reliability index following steps 2 to 4.The variance of reliability indexequals to 0.000072 after iterating 3 times.That meets the condition ofless than or equal to 0.0001,so the computation process terminated.The final computation result of reliability index is 4.147887 and the corresponding failure probability is 1.6778×10-5.

The failure probability of ring-stiffened cylindrical shell which involved to this paper is 1.3802×10-5when the uniform external load is 3.92 MPa[7].Compared with the result in this paper,the computational result of RSM-FORM is found accurate and this computational model can be applied in reliability analysis of ring-stiffened cylindrical shell structure.

4 Conclusions

According to characteristic of ring-stiffened cylindrical shell,the structural strength is analyzed by using the finite element methods.27 sets of valid data are selected and the failure state function is established.Then,FORM is improved by combining with RSM and the random variables are converted into independent and normalized parameters.The computational accuracy of RSM-FORM is improved greater than FORM because the quadratic term of the function is kept.At last,the reliability of ring-stiffened cylindrical shell is analyzed with RSM-FORM and the computational result of RSM-FORM is completely satisfied to the need of practical engineering.The research results and method of this paper may be used to reliability analysis and design of other structures.

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