CN102156819B - Pipeline crack equivalent stiffness calculation method based on stress intensity factor - Google Patents

Abstract

The invention discloses a method for calculating equivalent rigidity of pipeline cracks based on stress intensity factors. In this method, the pipe structure is discretized into a series of thin-walled rings nested in the radial direction, so as to obtain the stress intensity factor of the pipe structure, and then calculate the equivalent stiffness of the crack. In the study of the vibration-based crack diagnosis method, the local stiffness change of the structure caused by the crack can be described by the torsion wire spring model. Therefore, the obtained crack equivalent stiffness can be used to identify the location and size of the pipe crack. This method can obtain the equivalent stiffness of the crack at any position in the pipeline, avoiding a lot of tedious experimental work, the result is reliable, simple and easy to implement, and provides a basis for further diagnosis of pipeline cracks.

Description

Pipeline crack equivalent stiffness computing method based on stress intensity factor

Technical field

The invention belongs to the mechanical fault diagnosis field, be specifically related to a kind of pipeline crack equivalent stiffness computing method based on stress intensity factor.

Background technology

Pipeline is one of five large means of transports that comprise railway, highway, water transport, air transportation, occupies extremely important status in the productions such as petrochemical complex.Along with the continuous appearance of pipeline accident, the damage check of the security performance of pipeline is become present hot subject.The Dynamic Non-Destruction Measurement that is used at present piping system mainly contains Magnetic Flux Leakage Inspecting technology, ultrasonic detecting technology, eddy detection technology, acoustic emission and vibration detection etc.Vibration detection is obtained vibration parameters by pipeline configuration is carried out kinetic measurement, and with the change of the vibration parameters Main Basis as the recognition structure damage.Vibration detection is easy to use, only needs one-piece construction or partial structurtes are tested, and does not need pointwise to detect and just can determine pipeline configuration damage position and size.Therefore, method for detecting vibration can detect in-service industrial pipeline in a large number rapidly, and good application prospect is arranged.

In recent years, obtained many achievements based on the Method for Cracks of vibrating, in these researchs, crackle causes that the structure partial stiffness variation adopts the torsion line spring model to describe usually, thereby obtains the crackle equivalent stiffness by calculating stress strength factor.But wherein most research work concentrates in damage identification to the bar girder construction of Filled Rectangle cross section or circular section.Because pipeline configuration is not only equally bearing the extraneous load of various complexity with solid construction, and inside has the effect of stream (gas) body usually, the solid coupling of this stream causes the Crack Extension complexity of the hollow shaft type structures such as pipeline, the stress intensity factor dyscalculia, therefore relatively less based on the Crack Damage Study of recognition work of the hollow shaft type structures such as relevant pipeline of vibrating both at home and abroad.

Obtain the crackle equivalent stiffness for the difficulty that overcomes stress intensity factor calculating, people such as India scholar Maiti 2008 have proposed respectively two kinds of experimental techniques measuring based on quiet distortion and natural frequency in document " On prediction of crack in different orientations in pipe using frequency based approach ".Adopt experimental technique can obtain the crackle equivalent stiffness, but need a large amount of loaded down with trivial details experimental works of cost, nor may all test the crackle of all positions.Simultaneously, experimental technique also is unfavorable for disclosing the rule that the crackle equivalent stiffness changes.Therefore, the computing method of research crackle equivalent stiffness have great importance.

Summary of the invention

The object of the invention is to, a kind of pipeline crack equivalent stiffness computing method based on stress intensity factor are provided.The method by with pipeline configuration radially discrete be a series of nested thin-walled rings successively, utilize existing thin-walled ring stress intensity factor formula to try to achieve the stress intensity factor of pipeline, then calculate the equivalent stiffness of crackle.The method can be calculated the equivalent stiffness of optional position crackle in pipeline, has avoided a large amount of loaded down with trivial details experimental works, and reliable results is simple, for the diagnosis problem of further furtheing investigate pipeline crack provides the foundation.

To achieve these goals, the technical scheme taked of the present invention is:

1) with radially discrete, the successively nested thin-walled ring (number of thin-walled ring be made as n) identical for a series of wall thickness of pipeline, utilize discrete thin-walled ring structure parameter to try to achieve the stress intensity factor of each thin-walled ring according to thin-walled ring stress intensity factor formula, thereby obtain the stress intensity factor of whole pipeline;

2) increase the value of n, if the stress intensity factor precision of resulting each thin-walled ring satisfies the trueness error requirement, discrete value n satisfies the division requirement; If do not satisfy, continue to increase the n value, until be met the stress intensity factor of trueness error; Along with the continuous increase of n, the calculated value of stress intensity factor converges on exact solution;

3) utilize the stress intensity factor of n the thin-walled ring that obtains, by the strain energy of calculating pipeline, and then obtain the crackle equivalent stiffness;

4) will be met the crackle equivalent stiffness of accuracy requirement for diagnosis pipe vibration crackle.

The structure partial stiffness variation that the appearance of crackle can cause, thus the change of system vibration modal parameter (as natural frequency, the vibration shape etc.) caused.By seeking the relation of modal parameter and structural damage, can diagnose structural crack, thus position and the size of identification crackle.

Described with pipeline radially discrete be a series of nested thin-walled rings successively, thereby calculate the stress intensity factor of pipeline, comprise the following steps:

The inside and outside radius of pipeline is respectively R _a, R _b, to consider to have a transversal crack on pipeline, crack depth is h;

With pipeline radially evenly discrete be n thin-walled ring, by finding the solution the stress intensity factor of each thin-walled ring, can obtain the stress strength factor K of whole pipeline _I, and then try to achieve the crackle equivalent stiffness.The stress intensity factor of i thin-walled ring is designated as K _i, K _iCan calculate according to following formula:

$K_i={\mathrm{\σ}}_i\sqrt{R_i}{\left(\frac{\sqrt{2}}{{\mathrm{\ϵ}}_i}\right)}^{1/2}G\left(\mathrm{\θ}\right)$

In formula:

${\mathrm{\σ}}_i=M(I_i/I)/\left(\mathrm{\π}{R}_i^{2}t\right)$

${\mathrm{\ϵ}}_i^2=(t/R_i)/\sqrt{12(1-{\mathrm{\μ}}^2)}$

$G\left(\mathrm{\θ}\right)=\sin \mathrm{\θ}[1+\frac{1}{2}\frac{\mathrm{\θ}-\cot \mathrm{\θ}(1-\cot \mathrm{\θ})}{2\cot \mathrm{\θ}+\sqrt{2}\cot \left(\frac{\mathrm{\π}-\mathrm{\θ}}{\sqrt{2}}\right)}]$

T---the wall thickness of thin-walled ring;

R _i---the inside and outside radius mean value of i thin-walled ring;

θ---angle coordinate;

The moment of flexure at M---pipeline crack two ends;

I _i---the moment of inertia of i thin-walled ring xsect;

The moment of inertia of I---cross-section of pipeline;

When n satisfies

Condition the time, the solving precision of stress intensity factor satisfies trueness error, reaches≤1%.Along with the increase of discrete thin-walled number of rings amount n, precision can further improve.

The stress intensity factor of the pipeline that described utilization obtains by the strain energy of calculating pipeline, and then is tried to achieve the crackle equivalent stiffness, comprises the following steps:

The strain energy of i thin-walled ring is shown below:

$U_i=2{\∫}_0^{{\mathrm{\θ}}_i}{J}_i{R}_i\mathrm{td\θ}=2{\∫}_0^{{\mathrm{\θ}}_i}\frac{(1-{\mathrm{\μ}}^2){K_i}^2}{E}{R}_i\mathrm{td\θ}$

In formula:

θ _i---the open-angle of i thin-walled ring crack line;

${\mathrm{\θ}}_i=\arccos \left(\frac{R_b-h}{R_i}\right)$

J _i---the strain energy density function of i thin-walled ring.

The total strain energy of pipeline is:

$U=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{U}_i={\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="HDA0000057590420000021.png"\; state="\{\{state\}\}">M</figure-callout>}}^2\frac{2\sqrt{2}(1-{\mathrm{\&mu;}}^2)}{E{\mathrm{\&pi;}}^2{\mathrm{<figure-callout\; id="2"\; label="tI"\; filenames="HDA0000057590420000021.png"\; state="\{\{state\}\}">tI</figure-callout>}}^2}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\left(\frac{I_i^2}{R_i^2{\mathrm{\&epsiv;}}_i}{\&Integral;}_0^{{\mathrm{\&theta;}}_i}{G}^2\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}\right)$

Crackle added aspect of compliance c _ξWith crackle equivalent stiffness K _tFor:

$c_{\mathrm{\&xi;}}=\frac{{\&PartialD;}^{2}M}{\&PartialD;{\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="HDA0000057590420000021.png"\; state="\{\{state\}\}">M</figure-callout>}}^2}=\frac{4\sqrt{2}(1-{\mathrm{\&mu;}}^2)}{E{\mathrm{\&pi;}}^2{\mathrm{<figure-callout\; id="2"\; label="tI"\; filenames="HDA0000057590420000021.png"\; state="\{\{state\}\}">tI</figure-callout>}}^2}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\left(\frac{I_i^2}{R_i^2{\mathrm{\&epsiv;}}_i}{\&Integral;}_0^{{\mathrm{\&theta;}}_i}{G}^2\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}\right)$

K _t＝1/c _ξ。

The method of calculating pipeline crack equivalent stiffness provided by the present invention is compared with experimental technique, has following significant advantage:

1) can obtain the equivalent stiffness of optional position crackle in pipeline;

2) having avoided is to obtain a large amount of loaded down with trivial details experimental work that crackle pipe equivalent stiffness is carried out;

3) for further deeply diagnosing pipeline crack to provide the foundation.

Description of drawings

Fig. 1 is pipeline crack cross-sectional structure schematic diagram;

Fig. 2 is i thin-walled ring crack line cross-sectional structure schematic diagram;

Fig. 3 is the convergence schematic diagram of crackle calculating method of stiffness;

Fig. 4 takes the equivalent stiffness of the pipeline crack that the inventive method calculates to be used for diagnosis pipeline crack effect schematic diagram.

Embodiment

Below in conjunction with accompanying drawing, content of the present invention is described further:

With reference to shown in Figure 1, be the pipeline crack xsect, wherein crack depth h can also can be less than wall thickness greater than pipeline wall thickness, ξ, η, is two-dimensional coordinate system.

With reference to shown in Figure 2, be i thin-walled ring crack line xsect.Be n thin-walled ring because pipeline is radially discrete, can represent whole pipeline so n thin-walled ring combines.ξ, η, be two-dimensional coordinate system.

With reference to shown in Figure 3, be the convergence of crackle calculating method of stiffness.Pipeline is radially discrete is n thin-walled ring, and along with the continuous increase of n, the calculated value of crackle equivalent stiffness converges on exact solution.Horizontal ordinate crack depth in figure, ordinate represents the flexibility of crackle.

With reference to shown in Figure 4, for the equivalent stiffness that will take the pipeline crack that the inventive method calculates is used for the diagnosis pipeline crack.In figure, horizontal ordinate β represents the crackle relative depth, and ordinate α represents the relative depth of crackle.Be the level line of first three rank natural frequency in Fig. 4 (a), 4 (b), three position and degree of depth that isocontour intersection point has indicated crackle to exist.

The present invention implements according to the following steps:

1) with radially discrete, the successively nested thin-walled ring (number of thin-walled ring be made as n) identical for a series of wall thickness of pipeline, utilize discrete thin-walled ring structure parameter to try to achieve the stress intensity factor of each thin-walled ring according to thin-walled ring stress intensity factor formula, thereby obtain the stress intensity factor of whole pipeline;

2) increase the value of n, if the stress intensity factor precision of resulting each thin-walled ring satisfies the trueness error requirement, discrete value n satisfies the division requirement; If do not satisfy, continue to increase the n value, until be met the stress intensity factor of trueness error;

3) utilize the stress intensity factor of n the thin-walled ring that obtains, by the strain energy of calculating pipeline, and then obtain the crackle equivalent stiffness;

4) will be met the crackle equivalent stiffness of accuracy requirement for diagnosis pipe vibration crackle.The structure partial stiffness variation that the appearance of crackle can cause, thus the change of system vibration modal parameter (as natural frequency, the vibration shape etc.) caused.By seeking the relation of modal parameter and structural damage, can diagnose structural crack, thus position and the size of identification crackle.

Described with pipeline radially discrete be a series of nested thin-walled rings successively, thereby calculate the stress intensity factor of pipeline, comprise the following steps:

The inside and outside radius of supposing pipeline is respectively R _a, R _b, to consider to have a transversal crack on pipeline, crack depth is h, as shown in Figure 1.In order to calculate the crackle equivalent stiffness, at first the below calculates the stress intensity factor of pipeline.

With pipeline radially with wall thickness evenly discrete be n thin-walled ring, by finding the solution the stress intensity factor of each thin-walled ring, can obtain the stress strength factor K of whole pipeline _IThe crackle xsect of i thin-walled ring as shown in Figure 2.The stress intensity factor of i thin-walled ring is designated as K _i, calculate according to following formula:

$K_i={\mathrm{\&sigma;}}_i\sqrt{R_i}{\left(\frac{\sqrt{2}}{{\mathrm{\&epsiv;}}_i}\right)}^{1/2}G\left(\mathrm{\&theta;}\right)$

In formula:

${\mathrm{\&sigma;}}_i=M(I_i/I)/\left(\mathrm{\&pi;}{R}_i^{2}t\right)$

${\mathrm{\&epsiv;}}_i^2=(t/R_i)/\sqrt{12(1-{\mathrm{\&mu;}}^2)}$

$G\left(\mathrm{\&theta;}\right)=\sin \mathrm{\&theta;}[1+\frac{1}{2}\frac{\mathrm{\&theta;}-\cot \mathrm{\&theta;}(1-\cot \mathrm{\&theta;})}{2\cot \mathrm{\&theta;}+\sqrt{2}\cot \left(\frac{\mathrm{\&pi;}-\mathrm{\&theta;}}{\sqrt{2}}\right)}]$

T---the wall thickness of i thin-walled ring;

R _i---the inside and outside radius mean value of i thin-walled ring;

θ---angle coordinate;

The moment of flexure at M---pipeline crack two ends;

I _i---the moment of inertia of i thin-walled ring xsect;

The moment of inertia of I---cross-section of pipeline;

When n satisfies Condition the time, the solving precision of stress intensity factor satisfies trueness error, reaches≤1%.Along with the increase of discrete thin-walled number of rings amount n, raising can be advanced-be gone on foot to precision.

T and R _iComputing formula be:

(1) as h≤R _b-R _iThe time

t＝h/n

$R_i=(R_b-h)+t(i-\frac{1}{2}),$ i＝1，2，...n；

(2) as h＞R _b-R _iThe time,

t＝(R _b-R _a)/n

$r_i=R_a+t(i-\frac{1}{2})$ i＝1，2，...n；

After obtaining the stress intensity factor of pipeline, just can calculate the strain energy of pipeline, and then can be in the hope of the crackle equivalent stiffness according to Castigliano;

The strain energy of i thin-walled ring is shown below:

$U_i=2{\&Integral;}_0^{{\mathrm{\&theta;}}_i}{J}_i{R}_i\mathrm{td\&theta;}=2{\&Integral;}_0^{{\mathrm{\&theta;}}_i}\frac{(1-{\mathrm{\&mu;}}^2){K_i}^2}{E}{R}_i\mathrm{td\&theta;}$

In formula:

θ _i---the open-angle of i thin-walled ring crack line;

${\mathrm{\&theta;}}_i=\arccos \left(\frac{R_b-h}{R_i}\right)$

J _i---the strain energy density function of i thin-walled ring.

The total strain energy of pipeline is:

$U=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{U}_i={\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="HDA0000057590420000021.png"\; state="\{\{state\}\}">M</figure-callout>}}^2\frac{2\sqrt{2}(1-{\mathrm{\&mu;}}^2)}{E{\mathrm{\&pi;}}^2{\mathrm{<figure-callout\; id="2"\; label="tI"\; filenames="HDA0000057590420000021.png"\; state="\{\{state\}\}">tI</figure-callout>}}^2}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\left(\frac{I_i^2}{{\mathrm{<figure-callout\; id="2"\; label="R\; i"\; filenames="HDA0000057590420000021.png"\; state="\{\{state\}\}">R</figure-callout>}}_i^{}{\mathrm{\&epsiv;}}_i}{\&Integral;}_0^{{\mathrm{\&theta;}}_i}{G}^2\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}\right)$

According to Castigliano, crackle added aspect of compliance c _ξWith crackle equivalent stiffness K _tFor:

$c_{\mathrm{\&xi;}}=\frac{{\&PartialD;}^{2}M}{\&PartialD;{\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="HDA0000057590420000021.png"\; state="\{\{state\}\}">M</figure-callout>}}^2}=\frac{4\sqrt{2}(1-{\mathrm{\&mu;}}^2)}{E{\mathrm{\&pi;}}^2{\mathrm{<figure-callout\; id="2"\; label="tI"\; filenames="HDA0000057590420000021.png"\; state="\{\{state\}\}">tI</figure-callout>}}^2}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\left(\frac{I_i^2}{{\mathrm{<figure-callout\; id="2"\; label="R\; i"\; filenames="HDA0000057590420000021.png"\; state="\{\{state\}\}">R</figure-callout>}}_i^{}{\mathrm{\&epsiv;}}_i}{\&Integral;}_0^{{\mathrm{\&theta;}}_i}{G}^2\left(\mathrm{\&theta;}\right)\mathrm{d\&theta;}\right)$

K _t＝1/c _ξ。

Below by specific embodiment, the present invention is described in further details:

Embodiment 1:

The present embodiment is mainly verified the convergence of the method for calculating pipeline crack equivalent stiffness disclosed in this invention.The crackle pipe is divided into n thin-walled ring, and the parameter of pipeline is as follows: R _b=0.0189m, v=0.3, E=173.81GPa, R _a=0.0139m, ρ=7860kg/m ³

Due to equivalent stiffness numerical value when crack depth is smaller very large (crack depth is 0 o'clock, and equivalent stiffness is infinitely great) of crackle, so then decay fast is not directly perceived when adopting rigidity figure to study.Because flexibility and rigidity are reciprocal each other, therefore, adopt the flexibility figure of crackle to study at this.Pipeline is radially discrete is n thin-walled ring, and the value of n is taken as respectively 1,2,5,10,20.Along with the continuous increase of n, the calculated value of crackle equivalent stiffness converges on exact solution, as shown in Figure 3.Analysis result has been verified convergence.

Embodiment 2:

The present embodiment verifies that mainly the method for calculating pipeline crack equivalent stiffness disclosed in this invention is used for the validity of diagnosis pipeline crack.The crackle pipe is divided into n thin-walled ring, and the parameter of pipeline is as follows: R _b=0.0189m, v=0.3, E=173.81GPa, R _a=0.0139m, ρ=7860kg/m ³

The structure partial stiffness variation that the appearance of crackle can cause, thus the change of system frequency caused.By seeking position and the magnitude relationship of natural frequency and crackle, can diagnose structural crack.The present embodiment adopts the method disclosed in the present to calculate the equivalent stiffness of the pipeline crack of diverse location and size, and then tries to achieve first three rank natural frequency of system with respect to the Changing Pattern of the position of crackle and size.Then draw the level line of first three rank natural frequency, three position and degree of depth that isocontour intersection point has indicated crackle to exist are as table 1 and shown in Figure 4.Experiment show take the equivalent stiffness of the pipeline crack that the inventive method calculates to be used for the correctness of diagnosis pipe vibration crackle.

Table 1 crack position and depth recognition result

Claims (1)

1. The pipeline crack equivalent stiffness calculation method based on stress intensity factor, it is characterized in that, the method comprises the following steps: 1) The pipeline is discretized in the radial direction into a series of n thin-walled rings with the same wall thickness and nested in sequence, and the stress intensity factor of each thin-walled ring is obtained according to the thin-walled ring stress intensity factor formula by using the discrete thin-walled ring structure parameters; 2) Increase the value of n. If the accuracy of the stress intensity factor of each thin-walled ring meets the accuracy error requirements, the discrete value n meets the division requirements; if not, continue to increase the value of n until the stress that meets the accuracy error is obtained. strength factor; 3) Using the obtained stress intensity factors of the n thin-walled rings, the equivalent stiffness of the crack is obtained by calculating the strain energy of the pipeline; 4) The obtained crack equivalent stiffness that meets the accuracy requirements is used to diagnose pipeline cracks; The discretization of the pipeline radially into a series of thin-walled rings nested in sequence, so as to calculate the stress intensity factor of the pipeline, includes the following steps: Set the inner and outer radii of the pipeline as R _a and R _b respectively, the depth of the transverse crack on the pipeline as h, and the stress intensity factor K _I of the entire pipeline, and the stress intensity factor K _i of the i-th thin-walled ring is obtained as: ${\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{<span\; class="notranslate">\; (</span>\frac{\sqrt{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span>$ In the formula: ${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&pi;</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>$ ${\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\sqrt{\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}$ $\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; cot</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; cot</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; cot</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; +</span>\sqrt{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; cot</span>}<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&theta;</span>}}{\sqrt{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ]</span>$ t - the wall thickness of the thin-walled ring; R _i - the average value of the inner and outer radii of the i-th thin-walled ring; θ——angle coordinate; M—bending moment at both ends of the pipe crack; I _i ——moment of inertia of the cross-section of the i-th thin-walled ring; I——the moment of inertia of the pipe cross section; J _i ——the strain energy density function of the ith thin-walled ring; The method of determining the strain energy of the pipeline by using the obtained stress intensity factor of the pipeline, and then obtaining the equivalent stiffness of the crack comprises the following steps: The strain energy of the i-th thin-walled ring is as follows: ${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; J</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; td\&theta;</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}\frac{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>{{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; E.</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; td\&theta;</span>}$ In the formula: θ _i ——the opening angle of the i-th thin-walled annular crack; ${\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; arccos</span>}<span\; class="notranslate">\; (</span>\frac{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; h</span>}}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; )</span>$ The total strain energy of the pipe is: $\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\sqrt{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; E.</span>}{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; t</span>}{\mathrm{<span\; class="notranslate">\; I</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}<span\; class="notranslate">\; (</span>\frac{{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; G</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&theta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>$ In the formula: μ——Poisson's ratio; E - modulus of elasticity; The crack additional flexibility c _m and crack equivalent stiffness K _t are: ${\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; \&xi;</span>}}<span\; class="notranslate">\; =</span>\frac{{<span\; class="notranslate">\; \&PartialD;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; m</span>}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 4</span>}\sqrt{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; E.</span>}{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; tI</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}<span\; class="notranslate">\; (</span>\frac{{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; G</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&theta;</span>}<span\; class="notranslate">\; )</span>$ K _t = 1/c _ξ ; Discretizing the pipeline radially into a series of n thin-walled rings with the same wall thickness and nested in sequence, using the discrete thin-walled ring structure parameters to obtain the stress intensity factor of each thin-walled ring according to the stress intensity factor formula of the thin-walled ring, when n satisfies

When the conditions are met, the solution accuracy of the stress intensity factor meets the accuracy error and reaches ≤1%; The value of n is taken as 1, 2, 5, 10 or 20 respectively.

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