CN102567632B - Shore bridge structure wind vibration fatigue life forecasting method based on accumulated damage of probability - Google Patents

Abstract

A shore bridge structure wind vibration fatigue life forecasting method based on accumulated damage of probability includes the following steps: step 1 adopting a harmonic superposition method to simulate time domain waveform of wind load borne by a shore bridge structure and according with davenport power spectrum characteristics; step 2 enabling the wind load to be acted on a finite element model of the shore bridge structure, and calculating a stress response time interval of fatigue calculation points of the shore bridge structure; step 3 adopting a rain flow counting process to deal with the stress response time interval so as to obtain fatigue statistical characteristics of amplitude stress spectrum; and step 4 adopting a probability accumulated damage model and a method of a probability theory to forecast wind vibration fatigue life of the shore bridge structure at a certain degree of reliability. The shore bridge structure wind vibration fatigue life forecasting method has the advantages of applying to a complex shore bridge structure, and being wide in scope of application, high in calculating accuracy and capable of calculating reliability fatigue life of a wind resisting structure under the action of any random wind load.

Description

Bank bridge construction wind based on the accumulated damage of probability fatigue life forecasting method that shakes

Technical field

The wind that the present invention relates to a kind of bank bridge construction fatigue life forecasting method that shakes, is specifically related to a kind ofly can be applicable to complicated bank bridge construction, computational accuracy is high and can calculates the bank bridge construction wind based on accumulated damage of probability that the wind under a certain fiduciary level the shakes fatigue lifetime fatigue life forecasting method that shakes.

Background technology

Bank bridge crane works in seashore throughout the year, and sail-center is high, and front face area is large, and wind load effect long-term, that be interrupted can cause bank bridge metal construction to produce Cumulative Fatigue Damage.The fatigue life forecasting method that bank bridge construction wind shakes is a major issue in the design of bank bridge construction and working service, its design difficulty is the experiment on fatigue properties data of lack of materials and the probability model of Cumulative Fatigue Damage, and is difficult to forecast that the wind of bank bridge construction under a certain fiduciary level shakes fatigue lifetime.

For addressing the above problem, the people such as Deng Hongzhou are at < < civil engineering work journal > > 2003,36 (4): on 19-23, delivered " fatigue study of mast structure Wind induced Random " literary composition.This article calculates the stress response statistic of the tired calculation level of mast structure in frequency domain by random seismic response analysis method, the fatigue lifetime of calculating mast structure with Modified equivalent stress model and equivalent narrow band method respectively.Yet this life forecast method is only suitable in the forecast of fatigue lifetime of simple bank bridge wind resisting structure, and forecast precision is low, in addition, the Miner linear cumulative damage criterion of the method institute foundation does not reflect that fatigue failure is this fact of a random occurrence.

Summary of the invention

The object of the present invention is to provide a kind of bank bridge construction wind based on accumulated damage of probability fatigue life forecasting method that shakes, to solve the existing bank bridge construction wind fatigue life forecasting method that shakes, be only applicable to simple bank bridge construction and the low technical matters of forecast precision.

For achieving the above object, the invention provides a kind of bank bridge construction wind based on accumulated damage of probability fatigue life forecasting method that shakes, comprise the following steps:

The first step, adopts the suffered wind load time domain waveform that meets Davenport power spectrum characteristic of harmonic wave method of superposition simulation bank bridge construction;

Second step, in the finite element model of bank bridge construction, calculates the stress response time-histories of the tired calculation level of bank bridge construction by wind action;

The 3rd step, adopts rain flow method to process stress response time-histories, thereby obtains the tired statistical nature of luffing stress spectrum;

The 4th step, by accumulated damage of probability model, adopts the wind of probability theory method forecast bank bridge construction under a certain fiduciary level to shake fatigue lifetime.

According to the fatigue life forecasting method that shakes of the bank bridge construction wind based on accumulated damage of probability described in preferred embodiment of the present invention, its first step wind load time domain waveform simulation is further comprising the steps:

First, according to random vibration theory, set up the power spectral density of the wind speed matrix S (ω) of each wind action point of bank bridge construction that meets Davenport power spectrum characteristic, then according to the waveform method of superposition of following formula, calculate the time domain waveform of pulsation wind speed:

$v_j\left(t\right)=2\sqrt{\mathrm{\&Delta;\&omega;}}\underset{m=1}{\overset{j}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{N}{\mathrm{\&Sigma;}}}\left|H_{\mathrm{jm}}\left({\mathrm{\&omega;}}_{\mathrm{ml}}\right)\right|\cos ({\mathrm{\&omega;}}_{\mathrm{ml}}t+{\mathrm{\&phi;}}_{\mathrm{ml}})$ j＝1，2...n

Wherein, the application point number that n is wind load, v _j(t) be the pulsation wind speed of j wind action point, N is an abundant large positive integer, φ _mlfor be uniformly distributed in interval [0,2 π) random phase, Δ ω is defined as ω _uand ω _dbe respectively the upper and lower bound of band of interest, ω _mlfor two index frequencies, (l=1,2...N), H _jm(ω) be the element in the Cholesky split-matrix H (ω) of S (ω);

Afterwards, adopt the Bernoulli's theorem based in fluid mechanics, calculate wind load time domain waveform:

$P\left(t\right)=\frac{\mathrm{\&gamma;}}{2g}{\mathrm{\&mu;}}_{s}A{[\stackrel{\&OverBar;}{v}+v\left(t\right)]}^2$

Wherein, γ is air unit weight, and g is acceleration of gravity, μ _swith A be respectively the Shape Coefficient of structure and effective wind area, it is mean wind speed.

According to the fatigue life forecasting method that shakes of the bank bridge construction wind based on accumulated damage of probability described in preferred embodiment of the present invention, its finite element model is for adopting the finite element model of the complicated bank bridge construction of finite element software foundation, and it comprises beam element, bar unit and lumped mass unit.

According to the fatigue life forecasting method that shakes of the bank bridge construction wind based on accumulated damage of probability described in preferred embodiment of the present invention, the statistical study of its 3rd step luffing stress spectrum, the luffing stress spectrum that is the tired calculation level based on rain flow method opposite bank bridge construction is added up, obtain the cycle index of different stress amplitudes and different mean stresses, as the shake input of fatigue life prediction algorithm of wind.

According to the fatigue life forecasting method that shakes of the bank bridge construction wind based on accumulated damage of probability described in preferred embodiment of the present invention, its the 4th step accumulated damage of probability model, regard the fatigue failure of bank bridge as random occurrence, set up the computing formula of Cumulative Fatigue Damage stochastic variable D and fatigue of materials intensity stochastic variable K, method is determined the regularity of distribution of fatigue of materials intensity K by experiment, fatigue lifetime based on probability theory method forecast bank bridge under a certain fiduciary level, concrete D and the K that calculates according to the following formula tired calculation level

$D=\mathrm{\&Sigma;}{\left(\frac{{\mathrm{\&sigma;}}_{\mathrm{ai}}{\mathrm{\&sigma;}}_b}{{\mathrm{\&sigma;}}_b-{\mathrm{\&sigma;}}_{\mathrm{mi}}}\right)}^m{n}_i$ $K={\left(\frac{{\mathrm{\&sigma;}}_{\mathrm{aj}}{\mathrm{\&sigma;}}_b}{{\mathrm{\&sigma;}}_b-{\mathrm{\&sigma;}}_{\mathrm{mj}}}\right)}^m{N}_j$

Wherein, σ _bbe the strength degree of material, m is the constant relevant with material, stress ratio, load mode, σ _aiand σ _mirespectively stress amplitude and the mean stress of asymmetric stresses circulation, n _iand N _ibe respectively actual cycle number of times and the inefficacy cycle index of certain Cyclic Stress.Regard the fatigue failure of bank bridge as random occurrence, under different stress levels, carry out the sample value that fatigue experiment obtains fatigue of materials intensity K, by statistical study, set up the regularity of distribution of K, K obeys logarithm normal distribution

lnK～N(μ _k，σ _k)。

According to the fatigue life forecasting method that shakes of the bank bridge construction wind based on accumulated damage of probability described in preferred embodiment of the present invention, above-mentioned accumulated damage of probability model calculates the fatigue lifetime of bank bridge construction under fiduciary level R according to the following formula

$T\left(R\right)=\frac{\exp [{\mathrm{\&mu;}}_k+{\mathrm{\&sigma;}}_k{\mathrm{\&phi;}}^{-1}(1-R)}{\mathrm{\&Sigma;}{P}_i{\left(\frac{{\mathrm{\&sigma;}}_{\mathrm{ai}}{\mathrm{\&sigma;}}_b}{{\mathrm{\&sigma;}}_b-{\mathrm{\&sigma;}}_{\mathrm{mi}}}\right)}^m}$

Wherein, P _iσ _aistress-number of cycles proportion in total stress cycle index.

The present invention adopts the time domain waveform of the suffered wind load of harmonic wave method of superposition simulation bank bridge construction, utilize the stress response time-histories of the tired calculation level of ripe finite element software calculation of complex bank bridge construction, the cycle index of different stress amplitudes and mean stress in employing rain flow method statistics luffing stress spectrum, regard the fatigue failure of bank bridge construction as random occurrence, set up the probability model of Cumulative Fatigue Damage and fatigue strength, and the wind under a certain fiduciary level shakes fatigue lifetime based on probability theory method forecast bank bridge construction.Method of the present invention can be calculated the reliability fatigue life of wind resisting structure under any random wind.Meanwhile, adopt business finite element software to set up the finite element model of complicated bank bridge construction, the bank bridge construction finite element model of setting up like this not only can adapt to complicated bank bridge construction, can also reflect the dynamics of bank bridge construction, significantly improves computational accuracy.Therefore, compared with prior art, beneficial effect of the present invention is: be applicable to complicated bank bridge construction, and applied widely, computational accuracy is high and can calculate the reliability fatigue life of wind resisting structure under any random wind.

Accompanying drawing explanation

Fig. 1 is the shake process flow diagram of fatigue life forecasting method of the bank bridge construction wind that the present invention is based on accumulated damage of probability;

Fig. 2 be in the embodiment of the present invention bank bridge construction geometric model and and load point schematic diagram;

Fig. 3 is the rain-flow counting statistics schematic diagram of the stress response of tired calculation level in the embodiment of the present invention.

Embodiment

Below in conjunction with accompanying drawing and enumerate embodiment and illustrate the present invention.Following embodiment implements take technical solution of the present invention under prerequisite, provided detailed embodiment and concrete operating process, but protection scope of the present invention is not limited to following embodiment.

Refer to Fig. 1, a kind of bank bridge construction wind based on accumulated damage of probability fatigue life forecasting method that shakes, comprises the following steps:

S11: adopt the suffered wind load time domain waveform that meets Davenport power spectrum characteristic of harmonic wave method of superposition simulation bank bridge construction.

This step is further comprising the steps:

First, according to random vibration theory, set up the power spectral density of the wind speed matrix S (ω) of each wind action point of bank bridge construction that meets Davenport power spectrum characteristic, then according to the waveform method of superposition of following formula, calculate the time domain waveform of pulsation wind speed:

$v_j\left(t\right)=2\sqrt{\mathrm{\&Delta;\&omega;}}\underset{m=1}{\overset{j}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{N}{\mathrm{\&Sigma;}}}\left|H_{\mathrm{jm}}\left({\mathrm{\&omega;}}_{\mathrm{ml}}\right)\right|\cos ({\mathrm{\&omega;}}_{\mathrm{ml}}t+{\mathrm{\&phi;}}_{\mathrm{ml}})$ j＝1，2...n

Wherein, the application point number that n is wind load, v _j(t) be the pulsation wind speed of j wind action point, N is an abundant large positive integer, φ _mlfor be uniformly distributed in interval [0,2 π) random phase, Δ ω is defined as ω _uand ω _dbe respectively the upper and lower bound of band of interest, ω _mlfor two index frequencies, (l=1,2...N), H _jm(ω) be the element in the Cholesky split-matrix H (ω) of S (ω);

Afterwards, adopt the Bernoulli's theorem based in fluid mechanics, calculate wind load time domain waveform:

$P\left(t\right)=\frac{\mathrm{\&gamma;}}{2g}{\mathrm{\&mu;}}_{s}A{[\stackrel{\&OverBar;}{v}+v\left(t\right)]}^2$

Wherein, γ is air unit weight, and g is acceleration of gravity, μ _swith A be respectively the Shape Coefficient of structure and effective wind area, it is mean wind speed.

S12: wind action, in the finite element model of bank bridge construction, is calculated to the stress response time-histories of the tired calculation level of bank bridge construction.

Finite element model is for adopting the finite element model of the complicated bank bridge construction of business finite element software foundation, and it comprises beam element, bar unit and lumped mass unit.The bank bridge construction finite element model of setting up has like this reflected the dynamics of bank bridge construction, be applicable to complicated bank bridge construction, and computational accuracy is higher.In the finite element model of bank bridge construction, adopt Finite Element Method to obtain the stress response time-histories of the tired calculation level of bank bridge construction wind action.

S13: adopt rain flow method to process stress response time-histories, thereby obtain the tired statistical nature of luffing stress spectrum.

The luffing stress spectrum of the tired calculation level of this step based on rain flow method opposite bank bridge construction is added up, and obtains the cycle index of different stress amplitudes and different mean stresses, as the shake input of fatigue life prediction algorithm of wind.

S14:, by accumulated damage of probability model, adopt the wind of probability theory method forecast bank bridge construction under a certain fiduciary level to shake fatigue lifetime.

Accumulated damage of probability model, regard the fatigue failure of bank bridge as random occurrence, set up the computing formula of Cumulative Fatigue Damage stochastic variable D and fatigue of materials intensity stochastic variable K, method is determined the regularity of distribution of fatigue of materials intensity K by experiment, fatigue lifetime based on probability theory method forecast bank bridge under a certain fiduciary level, specifically regard the fatigue strength K of Cumulative Fatigue Damage D and material as stochastic variable, calculate according to the following formula D and the K of tired calculation level

$D=\mathrm{\&Sigma;}{\left(\frac{{\mathrm{\&sigma;}}_{\mathrm{ai}}{\mathrm{\&sigma;}}_b}{{\mathrm{\&sigma;}}_b-{\mathrm{\&sigma;}}_{\mathrm{mi}}}\right)}^m{n}_i$ $K={\left(\frac{{\mathrm{\&sigma;}}_{\mathrm{aj}}{\mathrm{\&sigma;}}_b}{{\mathrm{\&sigma;}}_b-{\mathrm{\&sigma;}}_{\mathrm{mj}}}\right)}^m{N}_j$

Wherein, σ _bbe the strength degree of material, m is the constant relevant with material, stress ratio, load mode, σ _aiand σ _mirespectively stress amplitude and the mean stress of asymmetric stresses circulation, n _iand N _ibe respectively actual cycle number of times and the inefficacy cycle index of certain Cyclic Stress.Regard the fatigue failure of bank bridge as random occurrence, under different stress levels, carry out the sample value that fatigue experiment obtains fatigue of materials intensity K, by statistical study, set up the regularity of distribution of K, K obeys logarithm normal distribution

lnK～N(μ _k，σ _k)。

Finally, above-mentioned accumulated damage of probability model calculates the fatigue lifetime of bank bridge construction under fiduciary level R according to the following formula:

$T\left(R\right)=\frac{\exp [{\mathrm{\&mu;}}_k+{\mathrm{\&sigma;}}_k{\mathrm{\&phi;}}^{-1}(1-R)}{\mathrm{\&Sigma;}{P}_i{\left(\frac{{\mathrm{\&sigma;}}_{\mathrm{ai}}{\mathrm{\&sigma;}}_b}{{\mathrm{\&sigma;}}_b-{\mathrm{\&sigma;}}_{\mathrm{mi}}}\right)}^m}$

Wherein, P _iσ _aistress-number of cycles proportion in total stress cycle index.

For understanding better technical scheme of the present invention, below provide an embodiment: certain type bank bridge construction during operation, in suffered wind load, mean wind speed is 15m/s, and pulsation wind speed meets Davenport power spectrum characteristic, uses the inventive method to forecast its reliability fatigue life.

(1) time domain waveform of simulated wind load

As shown in Figure 1, be the geometric model of certain type bank bridge construction.False wind load action is on the discrete node shown in Fig. 1, fluctuating wind meets Davenport power spectrum characteristic, set up the power spectral density of the wind speed matrix S (ω) of each wind action point of bank bridge construction, when mean wind speed equals 15m/s, according to harmonic wave method of superposition simulated wind pressure time-histories.Consider the natural mode of vibration of bank bridge, during simulated wind pressure time-histories, getting time step is 0.1s, T.T. length be 600s.

(2) the stress response time-domain analysis of finite element modeling and tired calculation level

Adopt NASTRAN software to set up the finite element model of the bridge construction of bank shown in Fig. 1, mainly adopt beam element, bar unit and lumped mass unit, totally 923 nodes, 978 unit.Blast time-histories is acted on the load point shown in Fig. 1, No. 479 nodes on triatic stay between No. 403 nodes on hound and doorframe of usining between No. 948 nodes on ladder frame rear pole, doorframe, as tired calculation level, adopt Finite Element Method to obtain the stress response time-histories of tired calculation level.

In the present embodiment, it should be noted that, the node numbering of the present embodiment is without optimization process, though have 923 nodes, node numbering is not from 1 to 923 serial number.Therefore, the node numbering of the present embodiment has more than and is limited to 1 to No. 923.

(3) rain flow method is processed stress response time-histories

Adopt rain flow method to analyze the stress response time-histories of each calculation level, obtain the frequency diagram of the stress amplitude shown in Fig. 2 and mean stress.

(4) reliable life of forecast bank bridge construction

Bank bridge is manufactured by steel Q345, and its material constant is m=7.806, σ _b=597.4MPa is μ by the distribution parameter that the fatigue experiment data under different stress levels obtain the fatigue strength stochastic variable of Q345 material in addition _k=55.3946, σ _k=0.32916.The accumulated damages of each tired calculation level in 600s such as calculating 403,479 and 948 are respectively 6.39 * 10 ¹⁴, 4.09 * 10 ¹⁶with 2.88 * 10 ¹⁶, by relatively finding that between doorframe, No. 479 nodes on triatic stay are dangerous point.Suppose the annual work of bank bridge 6300 hours, can obtain so the reliability fatigue life of structure under each fiduciary level and be respectively

The present invention adopts the time domain waveform of the suffered wind load of harmonic wave method of superposition simulation bank bridge construction, utilize the stress response time-histories of the tired calculation level of ripe finite element software calculation of complex bank bridge construction, the cycle index of different stress amplitudes and mean stress in employing rain flow method statistics luffing stress spectrum, regard the fatigue failure of bank bridge construction as random occurrence, set up the probability model of Cumulative Fatigue Damage and fatigue strength, and the wind under a certain fiduciary level shakes fatigue lifetime based on probability theory method forecast bank bridge construction.Method of the present invention can be calculated the reliability fatigue life of wind resisting structure under any random wind.Meanwhile, adopt business finite element software to set up the finite element model of complicated bank bridge construction, the bank bridge construction finite element model of setting up like this not only can adapt to complicated bank bridge construction, can also reflect the dynamics of bank bridge construction, significantly improves computational accuracy.Therefore, compared with prior art, beneficial effect of the present invention is: be applicable to complicated bank bridge construction, and applied widely, computational accuracy is high and can calculate the reliability fatigue life of wind resisting structure under any random wind.

The above, it is only better embodiment of the present invention, not the present invention is done to any pro forma restriction, any content that does not depart from technical solution of the present invention, any simple modification, equivalent variations and the modification above embodiment done according to technical spirit of the present invention, all belong to the scope of technical solution of the present invention.

Claims (5)

1. the fatigue life forecasting method that shakes of the bank bridge construction wind based on accumulated damage of probability, is characterized in that, comprises the following steps:

The first step, adopts the suffered wind load time domain waveform that meets Davenport power spectrum characteristic of harmonic wave method of superposition simulation bank bridge construction;

Second step, in the finite element model of bank bridge construction, calculates the stress response time-histories of the tired calculation level of bank bridge construction by wind action;

The 3rd step, adopts rain flow method to process stress response time-histories, thereby obtains the tired statistical nature of luffing stress spectrum;

The 4th step, by accumulated damage of probability model, adopt the wind of probability theory method forecast bank bridge construction under a certain fiduciary level to shake fatigue lifetime, described accumulated damage of probability model, regard the fatigue failure of bank bridge as random occurrence, set up the computing formula of Cumulative Fatigue Damage stochastic variable D and fatigue of materials intensity stochastic variable K, method is determined the regularity of distribution of fatigue of materials intensity K by experiment, fatigue lifetime based on probability theory method forecast bank bridge under a certain fiduciary level, concrete D and the K that calculates according to the following formula tired calculation level:

$\begin{array}{cc}D=\mathrm{\&Sigma;}{\left(\frac{{\mathrm{\&sigma;}}_{\mathrm{ai}}{\mathrm{\&sigma;}}_b}{{\mathrm{\&sigma;}}_b-{\mathrm{\&sigma;}}_{\mathrm{mi}}}\right)}^m{n}_i& K={\left(\frac{{\mathrm{\&sigma;}}_{\mathrm{aj}}{\mathrm{\&sigma;}}_b}{{\mathrm{\&sigma;}}_b-{\mathrm{\&sigma;}}_{\mathrm{mj}}}\right)}^m\end{array}{N}_j$

Wherein, σ _bbe the strength degree of material, m is the constant relevant with material, stress ratio, load mode, σ _aiand σ _mirespectively stress amplitude and the mean stress of asymmetric stresses circulation, n _iand N _ibe respectively actual cycle number of times and the inefficacy cycle index of certain Cyclic Stress, regard the fatigue failure of bank bridge as random occurrence, under different stress levels, carry out the sample value that fatigue experiment obtains fatigue of materials intensity K, by statistical study, set up the regularity of distribution of K, K obeys logarithm normal distribution lnK～N (μ _k, σ _k).

2. the bank bridge construction wind based on the accumulated damage of probability according to claim 1 fatigue life forecasting method that shakes, is characterized in that, the wind load time domain waveform simulation described in the first step is further comprising the steps:

First, according to random vibration theory, set up the power spectral density of the wind speed matrix S (ω) of each wind action point of bank bridge construction that meets Davenport power spectrum characteristic, then according to the waveform method of superposition of following formula, calculate the time domain waveform of pulsation wind speed:

$v_j\left(t\right)=2\sqrt{\mathrm{\&Delta;\&omega;}}\underset{m=1}{\overset{j}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{N}{\mathrm{\&Sigma;}}}\left|H_{\mathrm{jm}}\left({\mathrm{\&omega;}}_{\mathrm{ml}}\right)\right|\cos ({\mathrm{\&omega;}}_{\mathrm{ml}}t+{\mathrm{\&phi;}}_{\mathrm{ml}}),j=\mathrm{1,2}\&CenterDot;\&CenterDot;\&CenterDot;n$

Wherein, the application point number that n is wind load, v _j(t) be the pulsation wind speed of j wind action point, N is an abundant large positive integer, φ _mlfor be uniformly distributed in interval [0,2 π) random phase, Δ ω is defined as ω _uand ω _dbe respectively the upper and lower bound of band of interest, ω _mlfor two index frequencies, h _jm(ω) be the element in the Cholesky split-matrix H (ω) of S (ω);

Afterwards, adopt the Bernoulli's theorem based in fluid mechanics, calculate wind load time domain waveform:

$P\left(t\right)=\frac{\mathrm{\&gamma;}}{2g}{\mathrm{\&mu;}}_{s}A{[\stackrel{\&OverBar;}{v}+v\left(t\right)]}^2$

Wherein, γ is air unit weight, and g is acceleration of gravity, μ _swith A be respectively the Shape Coefficient of structure and effective wind area, it is mean wind speed.

3. the bank bridge construction wind based on the accumulated damage of probability according to claim 1 fatigue life forecasting method that shakes, it is characterized in that, described finite element model is for adopting the finite element model of the complicated bank bridge construction of finite element software foundation, and it comprises beam element, bar unit and lumped mass unit.

4. the bank bridge construction wind based on the accumulated damage of probability according to claim 1 fatigue life forecasting method that shakes, it is characterized in that, the statistical study of the luffing stress spectrum described in the 3rd step, the luffing stress spectrum that is the tired calculation level based on rain flow method opposite bank bridge construction is added up, obtain the cycle index of different stress amplitudes and different mean stresses, as the shake input of fatigue life prediction algorithm of wind.

5. the bank bridge construction wind based on the accumulated damage of probability according to claim 1 fatigue life forecasting method that shakes, is characterized in that, described accumulated damage of probability model calculates the fatigue lifetime of bank bridge construction under fiduciary level R according to the following formula

$T\left(R\right)=\frac{\exp [{\mathrm{\&mu;}}_k+{\mathrm{\&sigma;}}_k{\mathrm{\&phi;}}^{-1}(1-R)]}{\mathrm{\&Sigma;}{P}_i{\left(\frac{{\mathrm{\&sigma;}}_{\mathrm{ai}}{\mathrm{\&sigma;}}_b}{{\mathrm{\&sigma;}}_b-{\mathrm{\&sigma;}}_{\mathrm{mi}}}\right)}^m}$

Wherein, P _iσ _aistress-number of cycles proportion in total stress cycle index.