CN104833536A - Structure fatigue life calculation method based on non-linear cumulative damage theory - Google Patents

Abstract

A structure fatigue life calculation method based on the non-linear cumulative damage theory comprises the following steps: firstly, performing a stress analysis on a metal structure through the finite element technology and determining a large-stress-value area as a dangerous area, secondly, determining initial damage of the structure according to a served age and surface quality detection of the structure, thirdly, measuring a stress spectrum of the structure in a certain period through a stress detector, fourthly, calculating cumulative damage of the structure under the stress spectrum through the non-linear cumulative damage theory, and fifthly, calculating a fatigue life of the metal structure through the cumulative damage. According to the method, the multi-stage load loading sequence and an influence of the size of a load on the cumulative damage are taken into account, and precision of the fatigue life is higher than the precision of a fatigue life calculated according to the linear cumulative damage theory. The method only needs a small amount of material performance data such as a cyclic stress-strain curve and a constant-amplitude strain life curve, calculation is simple, and the method is suitable for engineering applications.

Description

A kind of structure fatigue life computing method based on Nonlinear Cumulative defect theory

Technical field

The invention provides a kind of structure fatigue life computing method based on Nonlinear Cumulative defect theory, belong to Analysis of Fatigue and reliability assessment technical field.

Background technology

According to statistics, the fatigue break that occurs in engineering reality destroys, and accounts for the 50%-90% of whole mechanical failure, is one of the most common form of machinery, structural failure.Product, under the anticorrosion ringing of load, can produce fatigue damage, and when impairment value reaches certain threshold value, structure is by fatigue failure.

Current variable amplitude loading calculates the more employing of damage and shows the linear rule of Palmgren-Miner, but its result of calculation has larger dispersion rate, and therefore life appraisal precision is lower.In recent years, some models based on various Theory of Fracture Mechanics and In-Situ Observation Technique are developed, consider the unrelieved stress and crack tip closing phenomenon that are caused by the material plastic deformation of crack tip, comprise some Nonlinear Cumulative defect theories based on experimental data.Although these model methods are more more accurate than Miner rule, obtain application to a certain extent, owing to needing a large amount of experimental datas to carry out matching, engineering adaptability is not strong.

Based on above background and analysis, the present invention proposes a kind of structure fatigue life computing method based on Nonlinear Cumulative defect theory, only need the material characteristic data that the cyclic stress-strain curve of material and normal width Strain life curves etc. are a small amount of, calculate simple, can in applied metal component in low circumferential strain analysis of Fatigue-life.

Summary of the invention

(1) object of the present invention: the feature of and former Nonlinear Cumulative defect theory computing method complexity low for Palmgren-Miner linear rule computational accuracy, there is provided a kind of Nonlinear Cumulative defect theory based on strain parameter of reduced form to calculate accumulated damage, thus calculate the fatigue lifetime of component.First it need the hazardous location determining component, then determines that initial damage substitutes into and calculate, obtain stress spectra by instrument or simulation means, by Nonlinear Cumulative defect theory, calculates the accumulated damage of metal construction under stress spectra and fatigue lifetime.

(2) technical scheme:

A kind of structure fatigue life computing method based on Nonlinear Cumulative defect theory that the present invention proposes, based on following hypothesis:

Suppose that 1 fatigue load spectrum is known, the load varied in size by m level forms, and every grade of damage caused is designated as D _m, D _mpermanent after producing exist;

Suppose that 2 work as D _maccumulation when reaching the limit values D, general D gets between 0.5-2, and the fatigue representing metal construction reaches capacity, and structural fatigue destroys.

The concrete steps of the method are as follows:

Step one, to be calculated by finite element analysis software or theory of mechanics, stress analysis is carried out to metal construction, determines that the region that in metal construction, stress value is larger is analyzed as hazardous location.

For finite element analysis software, the softwares such as NASTRAN, ADINA, ANSYS, ABAQUS, MARC, MAGSOFT, COSMOS can be utilized, set up the finite element model of metal construction, and the operating mode of getting one exemplary carries out statics and dynamic analysis, find out under typical condition, stress value or the larger dangerous point of stress amplitude or hazardous location.For general metal construction, its stress reaches more than 80% of its yield limit, belongs to more dangerous, should be classified as hazardous location.

Step 2, according to the Years Of Service of metal construction and appearance quality testing, determine its initial damage.

According to the Years Of Service of metal construction and its material stress-cycle index curve (S-N curve), can preliminary judgement its whether enter fatigue.Low-cycle fatigue cycle index may be defined as 10 ³-10 ⁵number of times, high cycle fatigue cycle index is greater than 10 ⁵number of times.When cycle index of being on active service is more than 10 ³time, initial damage can count damage.According to appearance quality testing, whether it has crackle to exist, and can judge the concrete numerical value of initial damage.When crackle is less than 0.2mm, initial damage can be defined as between 0.01-0.1.

Step 3, by structural stress detecting instrument, measure strain-time spectrum or the stress-time spectrum of structure in hazardous location in some cycles, obtain stress spectra as shown in Figure 1.Meanwhile, ADAMS software can be utilized, carry out dynamics simulation, obtain its simulated stress spectrum.

By on metallic structures, adopt foil gauge patch form, measure the strain-time history of works under various typical condition.For simple metal works, the real data that the data that can directly adopt foil gauge to measure strain as it.For welded structure, the stress of its welding toe directly can not measure acquisition, adopts the mode of multimetering, and indirect calculation obtains the stress of welding toe.As shown in Figure 2, the stress of welding toe adopts shown in formula 1 the measuring point figure of the welded stress paster of template.As shown in Figure 3, the stress of welding toe adopts shown in formula 2 and formula 3 the measuring point figure of the welded stress paster of cast.

${\mathrm{\σ}}_{\mathrm{hs}}=\frac{x_2\·{\mathrm{\σ}}_1-x_1\·{\mathrm{\σ}}_2}{x_2-x_1}---\left(1\right)$

Wherein, σ _hswelding toe focus stress, σ ₁, σ ₂that distance welding toe is apart from being x respectively ₁, x ₂surface stress, can be obtained by FEM (finite element) calculation or stress test.

In Fig. 3, $a=0.2\sqrt{\mathrm{rt}},b_1=b_2=0.65\sqrt{\mathrm{rt}},b_3=0.5\sqrt{\mathrm{RT}}$

Welding toe Stress calculation formula is:

${\mathrm{\σ}}_{\mathrm{bhs}}=\frac{b_1\·{\mathrm{\σ}}_a-a\·{\mathrm{\σ}}_{b_1}}{b_1-a}---\left(2\right)$

${\mathrm{\σ}}_{\mathrm{chs}}=\frac{b_3\·{\mathrm{\σ}}_a-a\·{\mathrm{\σ}}_{b_3}}{b_3-a}---\left(3\right)$

In formula (1), (2), σ _bhsfor the focus stress of web member welding toe, σ _chsfor the focus stress of chord member welding toe, being respectively distance toe of weld is a, b ₁, b ₃surface stress, the same Fig. 3 of other parameters.

Step 4, by Nonlinear Cumulative defect theory, calculate the accumulated damage of metal construction under stress spectra.By accumulated damage, calculate the fatigue lifetime of metal construction.When each circulation damage accumulation until lost efficacy time total impairment value D _tbe 1, the fatigue lifetime of the component for this reason of cycle index i now.The flow process of whole calculating as shown in Figure 4.

Suppose D _tall circulations damage D _isummation, shown in (4).

$D_T=\underset{i==1}{\overset{{2N}_T}{\mathrm{\Σ}}}{D}_i---\left(4\right)$

D in this definition _ibe damaged by i-th normalization oppositely caused, work as D _tlost efficacy when being 1.Normal width strain-life formula adopts Basquin-Manson-Coffin (BMC) formula, shown in (5).

$\frac{\mathrm{\Δ\ϵ}}{2}=\frac{{\mathrm{\Δ\ϵ}}_e}{2}+\frac{{\mathrm{\Δ\ϵ}}_p}{2}=\frac{{\mathrm{\σ}}_f^{\text{'}}}{E}{\left({2N}_f\right)}^b+{\mathrm{\ϵ}}_f^{\text{'}}{\left({2N}_f\right)}^c---\left(5\right)$

Δ ε/2 are half of range of strain, Δ ε _efor elastic strain, Δ ε _pfor moulding strain, 2N _ffor number of occurrence during fatigue failure.

As material be Q345B time, elastic modulus E=200741MPa, fatigue strength exponent b=-0.0943, fatigue ductility index c=-0.5395, fatigue strength coefficient σ _f'=947.1MPa, fatigue ductile coefficient ε _f, there is cyclic hardening characteristic '=0.4644.

Strain normal width, the Crack Extension under each circulation can use hyperbolic sine function representation.For crack size a, the spreading rate of crack tip is greater than 2N in cycle index _kafter, can be written as:

$\frac{\mathrm{da}}{\mathrm{dN}}\∝\sinh \left(\frac{{2N}_k}{{2N}_f}\mathrm{\ρ}\right)---\left(6\right)$

2N _fbe under given plastic strain amplitude, until circulation total degree when losing efficacy, ρ is a scale factor, and be used for suitably adjusting the size of damage increment, da/dN has just had correct funtcional relationship with strain amplitude. 2N _fderive from BMC formula, γ can use-c/2 ε ' _freplace.

Calculate damage with formula (6), the damage normalization that i-th circulation produces is expressed as follows,

$D_i=\frac{\underset{k=1}{\overset{N_T+1}{\mathrm{\Σ}}}\sinh \left(\frac{{2N}_k}{{2N}_f}\mathrm{\ρ}\right)-\underset{k=1}{\overset{N_T}{\mathrm{\Σ}}}\sinh \left(\frac{{2N}_k}{{2N}_f}\mathrm{\ρ}\right)}{\underset{j=1}{\overset{N_f}{\mathrm{\Σ}}}\sinh \left(\frac{{2N}_j}{{2N}_f}\mathrm{\ρ}\right)}\≈\frac{{\∫}_{N_T}^{N_T+1}\sinh \left(\frac{{2N}_k}{{2N}_f}\mathrm{\ρ}\right){\mathrm{dN}}_k}{{\∫}_1^{N_f}\sinh \left(\frac{{2N}_j}{{2N}_f}\mathrm{\ρ}\right){\mathrm{dN}}_j}---\left(7\right)$

N _tthe cycle index for reaching required for accumulated damage, D _tby normal width range of strain circulate the damage caused.Each circulation damage accumulation until lost efficacy time total impairment value be 1.

D _tcan know from formula (5), be increment damage D _iand, shown in (8).

$D_T=\frac{\underset{k=1}{\overset{N_T}{\mathrm{\Σ}}}\sinh \left(\frac{{2N}_k}{{2N}_f}\mathrm{\ρ}\right)}{\underset{j=1}{\overset{N_f}{\mathrm{\Σ}}}\sinh \left(\frac{{2N}_j}{{2N}_f}\mathrm{\ρ}\right)}\≈\frac{{\∫}_1^{N_T}\sinh \left(\frac{{2N}_k}{{2N}_f}\mathrm{\ρ}\right){\mathrm{dN}}_k}{{\∫}_1^{N_f}\sinh \left(\frac{{2N}_j}{{2N}_f}\mathrm{\ρ}\right){\mathrm{dN}}_j}---\left(8\right)$

Formula (6) can be separated and be:

${2N}_T=\frac{{2N}_f}{\mathrm{\ρ}}\×{\cosh }^{-1}(D_T(\cosh \left(\mathrm{\ρ}\right)-\left(\cosh \left(\frac{\mathrm{\ρ}}{{2N}_f}\right)\right)+\cosh \left(\frac{\mathrm{\ρ}}{{2N}_f}\right))---\left(9\right)$

Through above-mentioned analysis, formula (4)-(9) constitute the basic model of essentially nonlinear cumulative damage theory.

(3) advantage and effect: the invention provides a kind of structure fatigue life computing method based on Nonlinear Cumulative defect theory, its advantage is:

1) the present invention calculates compared with method fatigue lifetime with traditional based on linear cumulative damage law Palmgren-Miner rule, consider the impact of multi-level load loading sequence, magnitude of load, precision fatigue lifetime that the ratio of precision linear cumulative damage law of this method calculates wants high.

2), compared with the present invention calculates method fatigue lifetime with traditional Nonlinear Cumulative defect theory, the material characteristic data that this method only needs the cyclic stress-strain curve of material and normal width Strain life curves etc. a small amount of, calculates simple, is suitable for engineer applied.

3) contemplated by the invention the impact of different initial damages on fatigue lifetime, go for simple metal and welded structure, applicable surface is wider.

Accompanying drawing explanation

In order to make content of the present invention more easily be clearly understood, below according to specific embodiment also by reference to the accompanying drawings, the present invention is further detailed explanation, wherein:

Fig. 1 is the measured stress spectrum of cast weld assembly under six working cycle and the comparison diagram of simulated stress spectrum.

Fig. 2 is that the pressure detection point of the welded welding toe of template is arranged.

Fig. 3 is that the pressure detection point of the welded welding toe of cast is arranged.

Fig. 4 is the Nonlinear Cumulative damage life-span calculation process based on strain.

Fig. 5 is the truss arm figure of certain crane.

Fig. 6 is the cast welding piece dimensional drawing of hazardous location on certain crane truss arm.

Fig. 7 is the torture test figure of cast welding piece.

Fig. 8 is the error of calculation contrast table of fatigue lifetime when considering different initial damage

Label in accompanying drawing 7 is:

Be rigidly connected 1, horizontal actuator 2, cast test specimen 3, vertical actuator 4.

Embodiment

Below in conjunction with drawings and Examples, the present invention is described in further details.

The welding structural element of case selection shown in Fig. 5, is taken from a joint hazardous location of rubber tired crane truss arm.Its material parameter is: when material is Q345B, elastic modulus E=200741MPa, fatigue strength exponent b=-0.0943, fatigue ductility index c=-0.5395, fatigue strength coefficient σ _f'=947.1MPa, fatigue ductile coefficient ε _f, there is cyclic hardening characteristic '=0.4644.

By calculating its complete accumulated damage, determine its fatigue lifetime.Concrete steps are as follows:

Step one: by finite element analysis software, determines the tired hazardous location of component.By setting up finite element model to Fig. 5, in Ansys, carrying out two typical condition analyses, obtaining the point that multiple stress value is larger.Two typical conditions are in table 1, and the dangerous point obtained is in table 2.

Table 1 design condition table

The FEM (finite element) calculation stress amplitude of table 2 dangerous point

By table 2,7 dangerous points can be found.

Step 2, according to the Years Of Service of metal construction and appearance quality testing, determine its initial damage.

According to its Years Of Service, this equipment has been on active service 9 years, and according to annual work 200 days, work every day 8 hours, work per hour 30 minutes, a cycle index per minute calculated, then military service cycle index is 4.32 × 105 times, enters fatigue stage.Observe its welded cracks.At dangerous point 2, average crack length is approximately 3.5-5.5mm.At dangerous point 7, average crack length is approximately 3.5-6.0mm.On this jib, general admissible maximum crack length is 10-15mm.Therefore, this jib has had larger damage.Major part crackle has entered stable extension phase, more dangerous, and therefore jib needs to repair.Initial damage can list calculating in, between desirable 0.01-0.1.

Step 3, by structural stress detecting instrument, measure the strain-time spectrum of structure in hazardous location in some cycles, obtain stress spectra as shown in Figure 1.For simplicity, point layout is carried out in the place that have employed from toe of weld 30mm, using the data directly the measured stress value as dangerous point.Operating mode 1 time, made six working cycle, its stress spectra as shown in Figure 1.Utilize ADAMS software, the simulated stress spectrum of acquisition as shown in Figure 1.

Step 4, by Nonlinear Cumulative defect theory, calculate the accumulated damage of metal construction under stress spectra.

According to algorithm of the present invention, do following comparative analysis:

(1) when contrast initial damage is zero, the precision size of result of calculation;

(2) fatigue lifetime utilizing Palmgren-Miner linear approach to calculate is contrasted;

(3) result of theory calculate and experiment test is contrasted;

(4) error of measured stress spectrum and simulated stress spectrum is contrasted.

Table 3 is setting initial damage D _twhen=0, every error contrast fatigue lifetime.Can find out:

(1) error of the fatigue lifetime of being calculated by two kinds of stress spectra is no more than 10%, engineering demands.

(2) Nonlinear Cumulative defect theory is higher than the computational accuracy of linear cumulative damage law.Based on the computational accuracy of Nonlinear Calculation Method at about 1.14-1.19, and based on the computational accuracy of linear computational method at about 0.29-0.32, be tending towards very much conservative.

(3) initial damage has material impact to fatigue lifetime, nonlinear damage constitutive model is higher than the computational accuracy of Miner linear damage model, but also have gap from the average in torture test life-span, be tending towards slight danger, Here it is initial damage is on the impact in life-span.

Substituted into by initial damage and calculate, and to set initial damage be that 0.01-0.08 is analyzed, the torture test life-span of reference calculates according to simulated stress spectrum block.Its comparing calculation error result is in table 4 and Fig. 8.Can find out:

The initial damage of (1) two operating mode between 0.03 ~ 0.04, based on nonlinear Calculation of Fatigue Life error range within 10%.

(2) after initial damage setting is greater than 0.07, the error of calculation more than 20%, and will be tending towards conservative.

(3) consider from engineering, design needs certain surplus capacity, the initial damage of suggestion weld assembly can be decided to be between 0.04-0.07, while ensureing precision, has the conservative of appropriateness.

Table 3 result of calculation contrast (initial damage DT=0)

Table 4 is based on the error of calculation (the torture test life-span comes from simulated stress spectrum block) of Nonlinear Cumulative damage algorithm

Claims (1)

1. a fatigue life calculation method for Nonlinear Cumulative defect theory, is characterized in that:

It carries out under following assumed condition:

Suppose that 1 fatigue load spectrum is known, the load varied in size by m level forms, and every grade of damage caused is designated as D _m, D _mpermanent after producing exist;

Suppose that 2 work as D _maccumulation when reaching the limit values D, general D gets between 0.5-2, and the fatigue representing metal construction reaches capacity, and structural fatigue destroys;

The concrete steps of the method are as follows:

Step one, to be calculated by finite element analysis software or theory of mechanics, stress analysis is carried out to metal construction, determines that the region that in metal construction, stress value is larger is analyzed as hazardous location;

For finite element analysis software, the softwares such as NASTRAN, ADINA, ANSYS, ABAQUS, MARC, MAGSOFT, COSMOS can be utilized, set up the finite element model of metal construction, and the operating mode of getting one exemplary carries out statics and dynamic analysis, find out under typical condition, stress value or the larger dangerous point of stress amplitude or hazardous location; For general metal construction, its stress reaches more than 80% of its yield limit, belongs to more dangerous, should be classified as hazardous location;

Step 2, according to the Years Of Service of metal construction and appearance quality testing, determine its initial damage;

According to the Years Of Service of metal construction and its material stress-cycle index curve (S-N curve), can preliminary judgement its whether enter fatigue; Low-cycle fatigue cycle index may be defined as 10 ³-10 ⁵number of times, high cycle fatigue cycle index is greater than 10 ⁵number of times; When cycle index of being on active service is more than 10 ³time, initial damage can count damage; According to appearance quality testing, whether it has crackle to exist, and can judge the concrete numerical value of initial damage; When crackle is less than 0.2mm, initial damage can be defined as between 0.01-0.1;

Step 3, by structural stress detecting instrument, measure strain-time spectrum or the stress-time spectrum of structure in hazardous location in some cycles, obtain stress spectra as shown in Figure 1;

By on metallic structures, adopt foil gauge patch form, measure the strain-time history of works under various typical condition; For simple metal works, the real data that the data that can directly adopt foil gauge to measure strain as it; For welded structure, the stress of its welding toe directly can not measure acquisition, adopts the mode of multimetering, and indirect calculation obtains the stress of welding toe; As shown in Figure 2, the stress of welding toe adopts shown in formula 1 the measuring point figure of the welded stress paster of template; As shown in Figure 3, the stress of welding toe adopts shown in formula 2 and formula 3 the measuring point figure of the welded stress paster of cast;

${\mathrm{\σ}}_{\mathrm{hs}}=\frac{x_2\·{\mathrm{\σ}}_1-x_1\·{\mathrm{\σ}}_2}{x_2-x_1}---\left(1\right)$

Wherein, σ _hswelding toe focus stress, σ ₁, σ ₂that distance welding toe is apart from being x respectively ₁, x ₂surface stress, can be obtained by FEM (finite element) calculation or stress test;

In Fig. 3, $a=0.2\sqrt{\mathrm{rt}},b_1=b_2=0.65\sqrt{\mathrm{rt}},b_3=0.5\sqrt{\mathrm{RT}}$

Welding toe Stress calculation formula is:

${\mathrm{\σ}}_{\mathrm{bhs}}=\frac{b_1\·{\mathrm{\σ}}_a-a\·{\mathrm{\σ}}_{b_1}}{b_1-a}---\left(2\right)$

${\mathrm{\σ}}_{\mathrm{chs}}=\frac{b_3\·{\mathrm{\σ}}_a-a\·{\mathrm{\σ}}_{b_3}}{b_3-a}---\left(3\right)$

In formula (1), (2), σ _bhsfor the focus stress of web member welding toe, σ _chsfor the focus stress of chord member welding toe, being respectively distance toe of weld is a, b ₁, b ₃surface stress, the same Fig. 3 of other parameters;

Step 4, by Nonlinear Cumulative defect theory, calculate the accumulated damage of metal construction under stress spectra and fatigue lifetime;

Suppose D _tall circulations damage D _isummation, shown in (4).

$D_T=\underset{i==1}{\overset{{2N}_T}{\mathrm{\Σ}}}{D}_i---\left(4\right)$

D in this definition _ibe damaged by i-th normalization oppositely caused, work as D _tlost efficacy when being 1; Normal width strain-life formula adopts Basquin-Manson-Coffin (BMC) formula, shown in (5);

$\frac{\mathrm{\Δ\ϵ}}{2}=\frac{{\mathrm{\Δ\ϵ}}_e}{2}+\frac{{\mathrm{\Δ\ϵ}}_p}{2}=\frac{{\mathrm{\σ}}_f^{\text{'}}}{E}{\left({2N}_f\right)}^b+{\mathrm{\ϵ}}_f^{\text{'}}{\left({2N}_f\right)}^c---\left(5\right)$

Δ ε/2 are half of range of strain, Δ ε _efor elastic strain, Δ ε _pfor moulding strain, 2N _ffor number of occurrence during fatigue failure;

As material be Q345B time, elastic modulus E=200741MPa, fatigue strength exponent b=-0.0943, fatigue ductility index c=-0.5395, fatigue strength coefficient σ _f'=947.1MPa, fatigue ductile coefficient ε _f, there is cyclic hardening characteristic '=0.4644;

Strain normal width, the Crack Extension under each circulation can use hyperbolic sine function representation; For crack size a, the spreading rate of crack tip is greater than 2N in cycle index _kafter, can be written as:

$\frac{\mathrm{da}}{\mathrm{dN}}\∝\sinh \left(\frac{{2N}_k}{{2N}_f}\mathrm{\ρ}\right)---\left(6\right)$

2N _fbe under given plastic strain amplitude, until circulation total degree when losing efficacy, ρ is a scale factor, and be used for suitably adjusting the size of damage increment, da/dN has just had correct funtcional relationship with strain amplitude; 2N _fderive from BMC formula, γ can use-c/2 ε ' _freplace;

Calculate damage with formula (6), the damage normalization that i-th circulation produces is expressed as follows,

$D_i=\frac{\underset{k=1}{\overset{N_T+1}{\mathrm{\Σ}}}\sinh \left(\frac{{2N}_k}{{2N}_f}\mathrm{\ρ}\right)-\underset{k=1}{\overset{N_T}{\mathrm{\Σ}}}\sinh \left(\frac{{2N}_k}{{2N}_f}\mathrm{\ρ}\right)}{\underset{j=1}{\overset{N_f}{\mathrm{\Σ}}}\sinh \left(\frac{{2N}_j}{{2N}_f}\mathrm{\ρ}\right)}\≈\frac{{\∫}_{N_T}^{N_T+1}\sinh \left(\frac{{2N}_k}{{2N}_f}\mathrm{\ρ}\right){\mathrm{dN}}_k}{{\∫}_1^{N_f}\sinh \left(\frac{{2N}_j}{{2N}_f}\mathrm{\ρ}\right){\mathrm{dN}}_j}---\left(7\right)$

NT is the cycle index for reaching required for accumulated damage, D _tby normal width range of strain circulate the damage caused; Each circulation damage accumulation until lost efficacy time total impairment value be 1;

D _tcan know from formula (5), be increment damage D _iand, shown in (8);

$D_T=\frac{\underset{k=1}{\overset{N_T}{\mathrm{\Σ}}}\sinh \left(\frac{{2N}_k}{{2N}_f}\mathrm{\ρ}\right)}{\underset{j=1}{\overset{N_f}{\mathrm{\Σ}}}\sinh \left(\frac{{2N}_j}{{2N}_f}\mathrm{\ρ}\right)}\≈\frac{{\∫}_1^{N_T}\sinh \left(\frac{{2N}_k}{{2N}_f}\mathrm{\ρ}\right){\mathrm{dN}}_k}{{\∫}_1^{N_f}\sinh \left(\frac{{2N}_j}{{2N}_f}\mathrm{\ρ}\right){\mathrm{dN}}_j}---\left(8\right)$

Formula (6) can be separated and be:

${2N}_T=\frac{{2N}_f}{\mathrm{\ρ}}\×{\cosh }^{-1}(D_T(\cosh \left(\mathrm{\ρ}\right)-\left(\cosh \left(\frac{\mathrm{\ρ}}{{2N}_f}\right)\right)+\cosh \left(\frac{\mathrm{\ρ}}{{2N}_f}\right))---\left(9\right)$

Through above-mentioned analysis, formula (4)-(9) constitute the basic model of essentially nonlinear cumulative damage theory; By accumulated damage, calculate the fatigue lifetime of metal construction; When each circulation damage accumulation until lost efficacy time total impairment value D _tbe 1, the fatigue lifetime of the component for this reason of cycle index i now; The flow process of whole calculating as shown in Figure 4.