CN102567632A - 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

Based on the bank bridge construction wind of probability accumulated damage forecasting procedure fatigue lifetime of shaking

Technical field

The wind that the present invention relates to a kind of bank bridge construction forecasting procedure fatigue lifetime of shaking 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 the probability accumulated damage that the wind under a certain fiduciary level shakes fatigue lifetime forecasting procedure fatigue lifetime of shaking.

Background technology

Bank bridge crane works in the seashore throughout the year, and sail-center is high, and front face area is big, and wind load effect long-term, that be interrupted can cause bank bridge metal construction to produce Cumulative Fatigue Damage.Forecasting procedure fatigue lifetime that bank bridge construction wind shakes is a major issue in design of bank bridge construction and the 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, people such as Deng Hongzhou are at " civil engineering work journal " 2003,36 (4): 19-23 is last have been delivered " mast structure RANDOM WIND shake a fatigue study " literary composition.This article calculates the stress response statistic of the tired calculation level of mast structure with random seismic response analysis method in frequency domain, respectively with fatigue lifetime of Modified equivalent stress model and equivalent arrowband method calculating mast structure.Yet this life forecast method only is suitable for the forecast of fatigue lifetime of simple bank bridge wind resisting structure, and forecast precision is low, and in addition, the Miner linear cumulative damage criterion of this 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 forecasting procedure fatigue lifetime of shaking, only be applicable to simple bank bridge construction and the low technical matters of forecast precision to solve shake fatigue lifetime forecasting procedure of existing bank bridge construction wind based on the probability accumulated damage.

For achieving the above object, the present invention provides a kind of bank bridge construction wind based on probability accumulated damage forecasting procedure fatigue lifetime of shaking, and may further comprise the steps:

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

In second step,, calculate the stress response time-histories of the tired calculation level of bank bridge construction with the finite element model of wind action in the bank bridge construction;

In the 3rd step, adopt rain flow method to handle the stress response time-histories, thereby obtain the tired statistical nature of luffing stress spectrum;

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

According to the described bank bridge construction wind based on the probability accumulated damage of preferred embodiment of the present invention forecasting procedure fatigue lifetime of shaking, its first step wind load time domain waveform simulation further may further comprise the steps:

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

$v_j\left(t\right)=2\sqrt{\mathrm{\Δ\ω}}\underset{m=1}{\overset{j}{\mathrm{\Σ}}}\underset{l=1}{\overset{N}{\mathrm{\Σ}}}\left|H_{\mathrm{jm}}\left({\mathrm{\ω}}_{\mathrm{ml}}\right)\right|\cos ({\mathrm{\ω}}_{\mathrm{ml}}t+{\mathrm{\φ}}_{\mathrm{ml}})$ j＝1，2...n

Wherein, n is the application point number of wind load, v _j(t) be the pulsation wind speed of j wind action point, N is an abundant big positive integer, φ _MlFor be uniformly distributed in the 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 _MlBe two index frequencies,

(l=1,2...N), H _Jm(ω) be element among the Cholesky split-matrix H (ω) of S (ω);

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

$P\left(t\right)=\frac{\mathrm{\γ}}{2g}{\mathrm{\μ}}_{s}A{[\stackrel{\‾}{v}+v\left(t\right)]}^2$

Wherein, γ is an air unit weight, and g is an acceleration of gravity, μ _sWith A be respectively the build coefficient of structure and effective wind area,

It is mean wind speed.

According to the described bank bridge construction wind of preferred embodiment of the present invention forecasting procedure fatigue lifetime of shaking based on the probability accumulated damage; Its finite element model is the finite element model of the complicated bank bridge construction of employing finite element software foundation, and it comprises beam element, bar unit and lumped mass unit.

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

According to the described bank bridge construction wind of preferred embodiment of the present invention forecasting procedure fatigue lifetime of shaking based on the probability accumulated damage; Its 4th step probability accumulated damage 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; Confirm the regularity of distribution of fatigue of materials intensity K through experimental technique, based on the fatigue lifetime of probability theory method forecast bank bridge under a certain fiduciary level, specifically according to the D and the K of the tired calculation level of computes

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

Wherein, σ _bBe the strength degree of material, m is and material, stress ratio, the relevant constant of load mode, σ _AiAnd σ _MiBe respectively asymmetric stresses round-robin stress amplitude and mean stress, n _iAnd N _iBe respectively the 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, set up the regularity of distribution of K, K obeys logarithm normal distribution through statistical study

lnK～N(μ _k，σ _k)。

According to the described bank bridge construction wind based on the probability accumulated damage of preferred embodiment of the present invention forecasting procedure fatigue lifetime of shaking, above-mentioned probability accumulated damage model is according to the fatigue lifetime of computes bank bridge construction under reliability R

$T\left(R\right)=\frac{\exp [{\mathrm{\μ}}_k+{\mathrm{\σ}}_k{\mathrm{\φ}}^{-1}(1-R)}{\mathrm{\Σ}{P}_i{\left(\frac{{\mathrm{\σ}}_{\mathrm{ai}}{\mathrm{\σ}}_b}{{\mathrm{\σ}}_b-{\mathrm{\σ}}_{\mathrm{mi}}}\right)}^m}$

Wherein, P _iBe σ _AiStress-number of cycles proportion in the 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 the 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 shake fatigue lifetime based on the wind of probability theory method forecast bank bridge construction under a certain fiduciary level.Method of the present invention can be calculated the reliability fatigue life of wind resisting structure under any random wind.Simultaneously, adopt commercial 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, the scope of application is extensive, computational accuracy is high and can calculate the reliability fatigue life of wind resisting structure under any random wind.

Description of drawings

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

Fig. 2 is that bank bridge construction geometric model reaches and the load point synoptic diagram in the embodiment of the invention;

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

Embodiment

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

See also Fig. 1, a kind of bank bridge construction wind based on probability accumulated damage forecasting procedure fatigue lifetime of shaking may further comprise the steps:

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

This step further may further comprise the steps:

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

$v_j\left(t\right)=2\sqrt{\mathrm{\Δ\ω}}\underset{m=1}{\overset{j}{\mathrm{\Σ}}}\underset{l=1}{\overset{N}{\mathrm{\Σ}}}\left|H_{\mathrm{jm}}\left({\mathrm{\ω}}_{\mathrm{ml}}\right)\right|\cos ({\mathrm{\ω}}_{\mathrm{ml}}t+{\mathrm{\φ}}_{\mathrm{ml}})$ j＝1，2...n

Wherein, n is the application point number of wind load, v _j(t) be the pulsation wind speed of j wind action point, N is an abundant big positive integer, φ _MlFor be uniformly distributed in the 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 _MlBe two index frequencies,

(l=1,2...N), H _Jm(ω) be element among the Cholesky split-matrix H (ω) of S (ω);

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

$P\left(t\right)=\frac{\mathrm{\γ}}{2g}{\mathrm{\μ}}_{s}A{[\stackrel{\‾}{v}+v\left(t\right)]}^2$

Wherein, γ is an air unit weight, and g is an acceleration of gravity, μ _sWith A be respectively the build coefficient of structure and effective wind area,

It is mean wind speed.

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

Finite element model is the finite element model of the complicated bank bridge construction of the commercial finite element software foundation of employing, 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.With the finite element model of wind action, adopt Finite Element Method promptly to obtain the stress response time-histories of the tired calculation level of bank bridge construction in the bank bridge construction.

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

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

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

Probability accumulated damage 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, confirm the regularity of distribution of fatigue of materials intensity K, based on the fatigue lifetime of probability theory method forecast bank bridge under a certain fiduciary level through experimental technique; Specifically the fatigue strength K with Cumulative Fatigue Damage D and material regards stochastic variable as, according to the D and the K of the tired calculation level of computes

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

Wherein, σ _bBe the strength degree of material, m is and material, stress ratio, the relevant constant of load mode, σ _AiAnd σ _MiBe respectively asymmetric stresses round-robin stress amplitude and mean stress, n _iAnd N _iBe respectively the 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, set up the regularity of distribution of K, K obeys logarithm normal distribution through statistical study

lnK～N(μ _k，σ _k)。

At last, above-mentioned probability accumulated damage model is according to the fatigue lifetime of computes bank bridge construction under reliability R:

$T\left(R\right)=\frac{\exp [{\mathrm{\μ}}_k+{\mathrm{\σ}}_k{\mathrm{\φ}}^{-1}(1-R)}{\mathrm{\Σ}{P}_i{\left(\frac{{\mathrm{\σ}}_{\mathrm{ai}}{\mathrm{\σ}}_b}{{\mathrm{\σ}}_b-{\mathrm{\σ}}_{\mathrm{mi}}}\right)}^m}$

Wherein, P _iBe σ _AiStress-number of cycles proportion in the total stress cycle index.

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

(1) time domain waveform of simulation wind load

As shown in Figure 1, be the geometric model of certain type bank bridge construction.The false wind load action is on discrete node shown in Figure 1; Fluctuating wind meets the Davenport power spectrum characteristic; Set up the wind speed power spectrum density 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 the 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 bank bridge construction shown in Figure 1, mainly adopt beam element, bar unit and lumped mass unit, totally 923 nodes, 978 unit.The blast time-histories is acted on the load point shown in Figure 1; As tired calculation level, adopt Finite Element Method to obtain the stress response time-histories of tired calculation level with No. 479 nodes on the triatic stay between No. 403 nodes on the hound between No. 948 nodes on the ladder frame rear pole, doorframe and doorframe.

In the present embodiment, need to prove that the node numbering of 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 present embodiment has more than and is limited to 1 to No. 923.

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

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

(4) reliable life of forecast bank bridge construction

The bank bridge is made 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 the 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, through finding that relatively No. 479 nodes on the triatic stay are dangerous point between doorframe.Suppose the annual work of bank bridge 6300 hours, can obtain the reliability fatigue life of structure under each fiduciary level so 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 the 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 shake fatigue lifetime based on the wind of probability theory method forecast bank bridge construction under a certain fiduciary level.Method of the present invention can be calculated the reliability fatigue life of wind resisting structure under any random wind.Simultaneously, adopt commercial 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, the scope of application is extensive, computational accuracy is high and can calculate the reliability fatigue life of wind resisting structure under any random wind.

The above; It only is preferable embodiment of the present invention; Be not that the present invention is done any pro forma restriction; Any content that does not break away from technical scheme of the present invention according to any simple modification, equivalent variations and the modification that technical spirit of the present invention is done above embodiment, all belongs to the scope of technical scheme of the present invention.

Claims (6)

1. the bank bridge construction wind based on probability accumulated damage forecasting procedure fatigue lifetime of shaking is characterized in that, may further comprise the steps:

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

In second step,, calculate the stress response time-histories of the tired calculation level of bank bridge construction with the finite element model of wind action in the bank bridge construction;

In the 3rd step, adopt rain flow method to handle the stress response time-histories, thereby obtain the tired statistical nature of luffing stress spectrum;

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

2. the bank bridge construction wind based on the probability accumulated damage according to claim 1 forecasting procedure fatigue lifetime of shaking is characterized in that, the described wind load time domain waveform simulation of the first step further may further comprise the steps:

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

$v_j\left(t\right)=2\sqrt{\mathrm{\Δ\ω}}\underset{m=1}{\overset{j}{\mathrm{\Σ}}}\underset{l=1}{\overset{N}{\mathrm{\Σ}}}\left|H_{\mathrm{jm}}\left({\mathrm{\ω}}_{\mathrm{ml}}\right)\right|\cos ({\mathrm{\ω}}_{\mathrm{ml}}t+{\mathrm{\φ}}_{\mathrm{ml}})$ j＝1，2...n

Wherein, n is the application point number of wind load, v _j(t) be the pulsation wind speed of j wind action point, N is an abundant big positive integer, φ _MlFor be uniformly distributed in the 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 _MlBe two index frequencies,

(l=1,2...N), H _Jm(ω) be element among the Cholesky split-matrix H (ω) of S (ω);

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

$P\left(t\right)=\frac{\mathrm{\γ}}{2g}{\mathrm{\μ}}_{s}A{[\stackrel{\‾}{v}+v\left(t\right)]}^2$

Wherein, γ is an air unit weight, and g is an acceleration of gravity, μ _sWith A be respectively the build coefficient of structure and effective wind area,

It is mean wind speed.

3. the bank bridge construction wind based on the probability accumulated damage according to claim 1 forecasting procedure fatigue lifetime of shaking; It is characterized in that; Described finite element model is the finite element model of the complicated bank bridge construction of employing finite element software foundation, and it comprises beam element, bar unit and lumped mass unit.

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

5. the bank bridge construction wind based on the probability accumulated damage according to claim 1 forecasting procedure fatigue lifetime of shaking; It is characterized in that; Described probability accumulated damage model of the 4th step; 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, confirm the regularity of distribution of fatigue of materials intensity K through experimental technique; Based on the fatigue lifetime of probability theory method forecast bank bridge under a certain fiduciary level, specifically according to the D and the K of the tired calculation level of computes

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

Wherein, σ _bBe the strength degree of material, m is and material, stress ratio, the relevant constant of load mode, σ _AiAnd σ _MiBe respectively asymmetric stresses round-robin stress amplitude and mean stress, n _iAnd N _iBe respectively the 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, set up the regularity of distribution of K, K obeys logarithm normal distribution through statistical study

lnK～N(μ _k，σ _k)。

6. the bank bridge construction wind based on the probability accumulated damage according to claim 5 forecasting procedure fatigue lifetime of shaking is characterized in that, said probability accumulated damage model is according to the fatigue lifetime of computes bank bridge construction under reliability R

$T\left(R\right)=\frac{\exp [{\mathrm{\μ}}_k+{\mathrm{\σ}}_k{\mathrm{\φ}}^{-1}(1-R)}{\mathrm{\Σ}{P}_i{\left(\frac{{\mathrm{\σ}}_{\mathrm{ai}}{\mathrm{\σ}}_b}{{\mathrm{\σ}}_b-{\mathrm{\σ}}_{\mathrm{mi}}}\right)}^m}$

Wherein, P _iBe σ _AiStress-number of cycles proportion in the total stress cycle index.

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