nonlinear analysis method of high-strength steel based on

Research ArticleNonlinear Analysis Method of High-Strength Steel Based on LocalBuckling Fiber Hinge

Yuhua Wang and Hengchao Zheng

School of Electricity and Engineering Nanjing Institute of Railway Technology Nanjing Jiangsu 210031 China

Correspondence should be addressed to Hengchao Zheng zhenghengchaonjrtseducn

Received 24 February 2021 Revised 30 March 2021 Accepted 2 April 2021 Published 24 April 2021

Academic Editor Yi-Zhang Jiang

Copyright copy 2021 Yuhua Wang and Hengchao Zheng is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

Based on the analysis and summary of the research and application status of domestic and foreign high-strength steel localbuckling fiber hinged rod stability and ordinary steel local buckling fiber hinged rod stability control this paper proposes a fiberhinged rod suitable for spatial structures subject to local buckling e new type is high-strength steel composite local bucklingfiber hinged pressure rod e influence of related parameters in the dislocation string model and the dislocation couple modelon the ultrasonic nonlinear parameters is deeply analyzed And from the perspective of contact nonlinear acoustics themechanism of the ultrasonic nonlinear response of the crack is analyzed and the finite element software ABAQUS is used tosimulate it e relationship between the nonlinear parameter and the internal crack shape of the material is simulated andanalyzed which proves the nonlinearity A series of nonlinear ultrasonic testing was performed on three groups of FV520Bhigh-strength steel fatigue specimens using a nonlinear testing system Analyzing the results it is found that the material has agood ultrasonic nonlinear cumulative effect and the microcracks have a greater impact on the ultrasonic nonlinear responsee β-N curves under three sets of fatigue tests are obtained e results show that the nonlinear parameters are very sensitiveto the fatigue damage of FV520B high-strength steel and the ultrasonic nonlinear parameters generally increase with theincrease in the number of fatigue cycles

1 Introduction

High-strength steel refers to steel with a standard value of yieldstrength between 460 and 690MPa With the development ofnew steel production processes such as microalloying tech-nology and thermomechanical treatment technology high-strength steel especially compared to earlier high-strengthsteels has a higher cleanliness Nb V and Ti are used as theelements e representative microalloying replaces the tra-ditional carbon element strengthening method which not onlyimproves the yield strength of steel but also improves itsplasticity and toughness [1] e new high-strength steel de-veloped by this new process has the characteristics of highstrength good toughness and good processing and weldabilityand has been applied in many practical projects at home andabroad [2 3]

For high-strength steel welded I-shaped section mem-bers in order to make full use of the advantages of highstrength the section design can be more developed but thiscan easily exceed the current specification limits especiallythe height-to-thickness ratio of the web [4] Under the loadof the axial compression member with the web height-to-thickness ratio exceeding the limit the web buckles first butthe bearing capacity of the entire member does not reach themaximum and there is still a lot of room for ascent [5 6]erefore it is necessary to appropriately relax the height-to-thickness ratio limit of the web so that it can give full playto its own strength advantages and also improve the eco-nomics of welding I-shaped cross section members [7 8]However there are relatively few studies on high-strengthsteels with a yield strength of 690MPa and there are evenfewer design codes applicable to it e existing

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 5541772 12 pageshttpsdoiorg10115520215541772

specifications for high-strength steels simply apply designformulas suitable for ordinary-strength steels which greatlyrestricts the promotion and application of high-strengthsteel structures [9]

is paper proposes a simplified calculation model of thecomposite local buckling fiber hinged compression rod(using the BEAM188 element node coupling method) andthe correctness of the simplified calculation model is verifiedby comparison and analysis with solid elements In the initialdefect the residual stress will have an unfavorable effect onthe stable bearing capacity In this paper the analysis of thelocal buckling fiber hinged compression rod of high-strengthsteel with different slenderness ratios shows that the weldingresidual stress has little effect on the stable bearing capacity(when the residual stress is not considered the maximumdifference of the stable bearing capacity is only 302) Inthis paper the fabricated FV520B high-strength steel fatiguespecimen was tested by a nonlinear ultrasonic testing systemrough the data collection and processing the fundamentalwave and the second harmonic amplitude were extractede relationship between the ultrasonic nonlinear param-eters and the ultrasonic propagation distance in the materialwas studied e ultrasonic nonlinear response was detectedin the original plate specimen without fatigue damage in-dicating the inherent nonlinear characteristics of the ma-terial itself As the propagation distance increases theultrasonic nonlinear coefficient increases approximatelylinearly indicating that the nonlinear response has a cu-mulative effect Testing the notched samples that cycle for acertain number of cycles also found the cumulative effect ofthe ultrasonic nonlinear response e β-N curves ofFV520B material under three stress levels are obtained eresults show that the nonlinear parameters are very sensitiveto the early fatigue damage of FV520B high-strength steele nonlinear parameters show an increasing trend with theincrease of the number of fatigue cycles e relationshipbetween the ultrasonic nonlinear parameters of FV520Bhigh-strength steel and the fatigue cycle cycles can be used tocharacterize the early fatigue damage degree of the materialand reflect its fatigue life

2 Related Work

At present there are still few domestic studies on the overallstability and local stability of high-strength steel beams [10]Relevant scholars have studied the stability of high-strengthsteel thin-walled box-section beams mainly analyzing theeffects of the slenderness ratio flange width-to-thicknessratio and section side-length ratio of the box beam on itsoverall stability [11] e steel studied is 6mm thick18Mn2Cr Mo BA high-strength steel with a yield strength of745MPa is type of steel is generally used in bridge en-gineering Relevant scholars pointed out that the slendernessratio has little effect on the ultimate load of a box-shapedbeam in a purely bending state [12] After changing theslenderness ratio of the beam the ultimate load changeswithin 5 e flange width of the beam is relatively smallAfter the load-displacement curve appears at the highestpoint the curve slowly and steadily declines when the flange

width-to-thickness ratio of the beam is increased the flangeor web has already experienced local instability after the loadis added to the ultimate load [13] e elastic sectionmodulus of the beam decreases so the slope of thedescending section after the highest point of the load-dis-placement curve increases significantly It can be seen thatthe width-to-thickness ratio of the flange affects the overallstability of the beam to a greater extent When the cross-sectional side length ratio of the beam increases from 30 to60 the web height-to-thickness ratio continues to increaseand local instability is prone to occur in the compressionzone of the web which leads to an acceleration of the overallstability of the ultimate load drop [14]

e research on the local-total related buckling of box-shaped cross section members has a long history Domesticand foreign researches mainly focus on the local-total relatedbuckling of ordinary steel members e research on thelocal-total related buckling of high-strength steel membersmainly focuses on the axis Relevant scholars have usednumerical analysis methods to study the relative bucklingultimate bearing capacity of ordinary steel members withthin-walled box-shaped sections under axial compressionwith hinged ends [15] Research has found that when theultimate bearing capacity of the component is reached thebearing capacity of the component composed of weak panelswill decrease sharply [16] e researchers used the testresults of four axial compression members with a nominalyield strength of 390MPa to compare with the finite elementmodel [17] e research shows that residual stress and localoverall geometric defects have a great influence on the ul-timate bearing capacity of members Researchers haveproposed a formula for calculating the local-to-whole rel-ative buckling ultimate bearing capacity of axial compres-sion members with large width-to-thickness ratio based onthe effective width method [18] Relevant scholars haveconducted experimental research on three S355 steel weldedthin-walled box-shaped eccentric components established afinite element model considering the effects of initial geo-metric defects and residual stress and compared the ex-perimental results and they found that the two are in goodagreement Finally the verified finite element model is usedto study the influence of initial defects on the ultimatebearing capacity of the component [19]

Relevant scholars have respectively measured and ana-lyzed the overall geometric initial defects of I-shaped sectionmembers with nominal yield strengths of 460MPa and960MPa [20]e results show that themeasured amplitude ofthe overall geometric initial defects of most high-strength steelsis less than 11000e researchers measured and analyzed theoverall geometric initial defects of the I-shaped sectionmembers with nominal yield strengths of 460MPa 550MPaand 690MPa and the results showed that the maximumamplitudes were 1541 1339 and 1702 [21] Relevant scholarshave studied the effect of geometric initial defects on thestability of ordinary steel and high-strength steel [22] eresults show that compared with ordinary steel membersgeometric initial defects have less effect on the stability coef-ficient of high-strength steel members compared with theultimate bearing capacity the local buckling capacity is more

2 Mathematical Problems in Engineering

sensitive to local geometric initial imperfections Researchershave conducted experimental research and numerical analysison the overall stability and local stability of high-strength steelaxial compression members and have accumulated a wealth ofexperimental data and important research results [23] Inparticular the residual stress distribution of high-strength steelwelded components is measured by the cutting method andthe residual stress distribution model suitable for different steelstrength grades is summarized which provides reliable residualstress treatment for the future finite element analysis of high-strength steel welded components method Relevant scholarsuse ANSYS finite element software to simulate the axialcompression components of hot-rolled high-strength equi-lateral angle steel Studies have shown that the initial defectshave a smaller effect on the local stability bearing capacity ofhigh-strength hot-rolled equilateral angle steel axial com-pression members than ordinary steel axial compressionmembers e researchers analyzed and studied the ultimatebearing capacity of Q345 Q390 and Q420 steel welded box-section axial compression members whose aspect ratio exceedsthe limit e calculation results are compared with the directstrength method and the effective yield strength method estudy shows that the stability coefficient should be checkedaccording to the type a section in the 03 specification the directstrength method and the finite element calculation result are ingood agreement while the effective yield strength method isslightly conservative

3 Steel Tube Buckling Theory and FiniteElement Simulation Method

31 e Basic eory of Axial Buckling According to thedeformation state of the member during buckling it can bedivided into global buckling local buckling and global-localrelated buckling When the component is only buckled as awhole its parts will not be severely deformed and only thewhole body will undergo a huge change in configurationbefore and after buckling local buckling means that afterbuckling of the component only the local elements willdeform significantly related buckling is a coupling of globalbuckling and local buckling that is when buckling occursboth the overall and local elements of the member undergo alarge configuration change Due to the inevitable existence oflocal and overall defects related buckling is very common inactual engineering and should be paid attention to

As shown in Figure 1 a cylindrical shell subjected toaxial pressure can only maintain a balanced state underconditions far below the critical buckling load after localbuckling Its load-displacement curve is shown in Figure 1OABrsquo belongs to unstable bifurcation instability and thisform of buckling is also called limited interference bucklingUnder minimal external interference the cylindrical shellmay jump from the stable equilibrium state before bucklingto the nonadjacent equilibrium state before reaching thecritical load e path is shown in the curve OArsquoCB inFigure 1 It should be noted that geometric defects have agreat influence on this type of member making its actualultimate bearing capacity far less than the elastic criticalbuckling load Pcr

32 Calculation Method of Elastic Buckling Problem ecalculation of structural buckling is a complex and tediousprocess which involves a series of geometric parameters andinitial conditions of the component such as the structuralsystem the length of the component the connection conditionthe section form the size of the initial defect and the residualcaused by the processing andmanufacturing of the component

e static balance method is one of the most importantmethods to solve the elastic stability limit load of thestructure According to the force situation of the structurewith small deformation after the buckling of the memberthe buckling load is solved by the mechanical balanceequation Usually more than one buckling load is obtainedaccording to this method and the minimum value should betaken as the true buckling load of the member e balancemethod fully satisfies the mechanical equilibrium state of thecomponent after buckling so its calculation result is anaccurate value However it is difficult to find that the balancemethod is only suitable for simple mechanical models andsingle boundary conditions It is generally difficult to obtainthe buckling load of components with various forms ofmidsections and complex forces and boundary conditionsusing the balance method

e emergence of the energy method can solve theproblem of elastic stability with complex force conditions orstructures with diverse forms and compositions When thestructure is subjected to conservative forces the total po-tential energy function can be established according to itsdeformed state At this time the total potential energyshould include the potential energy caused by the externalload of the component and the strain energy caused by thedeformation of the internal structure According to theprinciple of standing value of potential energy the potentialenergy has a standing value when the component is inequilibrium so the buckling load solution is carried out intwo steps Firstly the balance equation is obtained by takingthe first-order variation of the total potential energy to thedisplacement and then the bifurcation buckling load isobtained by solving the balance equation Different from thebalance method the buckling load obtained by the energymethod is generally only an approximate solution but if thestructure deformation form and displacement function canbe determined in advance the exact solution can be ob-tained It is worth mentioning that the energy method canjudge the stability of bifurcation buckling according to thepositive and negative values of the second-order variation ofthe total potential energy that is when the second-ordervariation is negative the structure is unstable bifurcationbuckling

4 Theoretical Analysis of the Stability of AxiallyCompressed Cylindrical Shells

41 eoretical Analysis of Linear Small Deflection For thelinear small deflection theory analysis can be made from twoaspects axisymmetric and nonaxisymmetric When the axialcompression load reaches the critical buckling stress the steelpipe member may undergo symmetrical deformation andfailure of the steel pipe wall ldquoconvexrdquo or ldquoconcaverdquo

Mathematical Problems in Engineering 3

According to the theory of linear small deflectionbuckling the buckling equilibrium differential equation of acylindrical shell bearing uniform axial pressure is

wGnabla8 +tEz

4w

rzx4 + Nxnabla

4z2w

zx2 0 (1)

W is the displacement perpendicular to the shell surfaceG is the bending stiffness of the unit width bridge r and t arethe radius and thickness of the shell and Nx is the pressureon the unit arc length

We take the buckling deformation displacement func-tion as

w w1 sinπlx

x middot cosπly

y (2)

In the formula lx and ly are the buckling half-wavelengths in the x and y directions respectively After finishingthe critical buckling stress can be obtained

Nx Gπ2ψ minustE

r2π2ψ

(3)

Among them

ψ 1l2x

middot 1 minusl2xl2y

⎛⎝ ⎞⎠

2

(4)

e minimum value of Nx is

Nx t middot E

r middot3 v

2minus 11113872 1113873

1113969 (5)

Taking steel Poissonrsquos ratio 03 then

Nx 0582 middottE

r (6)

In actual engineering problems there are unavoidablegeometric initial defects in the components and the shell ismore sensitive to defects erefore the ultimate stress of acylindrical shell in experiments or actual engineering isalways much lower than the elastic critical buckling stress ofa cylindrical short column under axial compression In orderto explore the ultimate bearing capacity of cylindrical shellswith geometric initial imperfections it is necessary to adoptthe buckling analysis theory considering the nonlinear term

42 eoretical Analysis of Nonlinear Large Deflectione nonlinear large deflection buckling theory is proposed onthe basis of the linear small deflection theory ere is stillanother postbuckling equilibrium configuration far below thelinear critical load value and this postbuckling configuration iscloser to the real experimental phenomenon In this study onlythe calculation formula for the elastic critical buckling stress of ashort cylindrical tube under axial compression obtained by thesmall deflection theory is considered so the detailed calculationmethod of the large deflection buckling theory will not berepeated

5 Main Factors Affecting Local Buckling ofCylindrical Shells under Axial Compression

51 Influence of BoundaryConditions When the two ends ofthe member are unconstrained and can move freely in thecircumferential direction the critical buckling load will besignificantly reduced However in actual engineeringstructures the ends of circular pipe members usually haveconstrained members such as gusset plates and end bearingplates e end can be freely deformed rarely erefore forthe local buckling of the axially compressed cylindrical shortcolumn the influence of the component due to the boundaryconditions can be ignored

52 Influence of Residual Stress Welding is a physicalprocess of local heating and then cooling of componentmaterials Due to the uneven temperature field of the heatingarea and surrounding materials the welding area will de-form unevenly during the cooling process and cause variouswelding residual stresses erefore the residual stress is anunavoidable unfavorable factor for welded pipe membersIn-depth tests and numerical analysis can be done on theresidual stress distribution pattern of welded circular pipecomponents and its influence on the ultimate bearing ca-pacity of the components For ordinary-strength steel theresidual stress has a very obvious influence on the overalland local stable bearing capacity of the axially compressedround tube short column while for the high-strength steelround tube member with higher yield strength its distri-bution mode has a greater influence on the stable bearingcapacity of the high-strength steel

53 Parametric Finite Element Modeling Analysis Finiteelement modeling usually requires a lot of time for users andthe modeling process is cumbersome and error-prone esecondary development option based on the Python pro-gramming language in ABAQUS provides great conveniencefor usersrsquo parametric finite element modelinge secondarydevelopment of ABAQUS is mainly divided into two partssubprogram development and user graphical interfaceprogram development e development of core calculationsubprograms is based on the Fortran language Users can usethe interface to write their own material constitutive modelsaccording to different needs and can create new unit typesthe user graphical interface (GUI) program is a secondarydevelopment platform based on the Python language andusers can use Python language to write programs based onspecific modeling needs It can be expanded and can in-dependently develop pre- and postprocessing modules andGUI tools to realize a series of parametric modelingprocesses

e main research content of this subject is the localstability of high-strength steel tube axially compressed shortcolumns It is necessary to build more than 300 finite ele-ment short column models on the basis of verified finiteelement modelse number of models is large the repeated


operations are many and the parameters are met ereforein order to avoid errors in the manual modeling process andimprove the efficiency of finite element analysis this paperuses the Python language to write the corresponding script(Script) file and uses the ABAQUS environment to run thescript to realize the batch creation modification and sub-mission of the finite element model e main content in-cludes component generation material attributeassignment analysis step establishment interaction defini-tion meshing inp file modification etc

e parametric modeling process of the elastoplasticbuckling problem of cylindrical short columns under axialcompression is mainly divided into two steps namely thefirst-order linear elastic buckling modal analysis (buckle)and the second-order nonlinear elastoplastic bucklinganalysis e key to buckling analysis lies in the applicationof geometric initial imperfections as shown in Figure 2 InABAQUS the initial geometric defects cannot be introducedthrough the user graphical interfaceerefore the key pointto realize the parametric finite element analysis is to processthe inp file in batches and add keywords in the specified lineto introduce the initial geometric defects of the component

6 Structure and Calculation Model of High-Strength Steel Composite Local BucklingFiber Hinged Compression Rod

61 Flange Core-Casing High-Strength Steel Composite LocalBuckling Fiber Hinge Structure e buckling load of theaxial compression member is determined by the restraint ofthe end the calculated length factor the section charac-teristics and the material characteristics In this section theaddition of a hoop-type outer tube device provides effectiveconstraints on the hinged core steel tube at both ends toreduce the effective calculation length thereby increasingthe buckling load of the core steel tube Figure 3 shows thetheoretical buckling mode of the inner core steel pipe afteradding constraints

Flange-type core-sleeve combination local buckling fiberhinged rods include high-strength steel core steel pipesordinary steel outer sleeves flange discs connecting boltsrubber ring blocks and rubber gaskets e number of outercasing pipes is generally 4 (if the slenderness ratio of theinner steel pipe is less than 80 it can be reduced to 2 if theslenderness ratio of the inner steel pipe is greater than 150 itcan be increased to 6)e two outer sleeves at the upper andlower ends are welded with flange discs at only one endwhile the two outer sleeves in the middle are welded withflange discs at both ends the inner diameter of the flangedisc is the outer diameter of the core steel pipe and the outerflanges of the sleeve bolt holes are arranged at equal intervalsalong the periphery of the disk e inner core steel pipe isthe main compression component and a flange disc iswelded in its middle part and its outer diameter is the sameas the outer diameter of the outer pipe flange e inner coresteel pipe and the outer sleeve and the outer sleeve areconnected as a whole by a flange disc and high-strengthbolts Rubber round gaskets are added between each flange

disc to play a sealing role You fill rubber ring blocks at bothends of the outer tube and the corresponding positions of theinner core steel tube to seal and fix the inner core steel tubeere is a gap between the outer casing and the inner coresteel tube so that the weight of the composite component islighter and the inner core steel tube can freely undergo axialdeformation which plays a major role in resisting the axialpressure

e outer tube 2 of ordinary strength grade steel does notparticipate in the force and only serves as a restraint enumber of outer tubes is generally 4 (if the core steel tubeslenderness ratio is less than 80 it can be reduced to 2 if thecore steel tube slenderness ratio is greater than 150 it can beincreased to 6) e two outer sleeves at the upper and lowerends are welded with flange disc 3 only at one end and thetwo outer sleeves at the middle are welded with flange disc 3at both ends e inner diameter of the flange disc 3 is theouter diameter of the core steel pipe and four bolt holes arearranged at equal intervals along the periphery e innercore steel pipe and the outer sleeve and the outer sleeve areconnected as a whole through a flange disc and high-strength bolts ere is a gap between the outer casing andthe inner core steel tube so that the weight of the compositemember is lighter and the inner core steel tube can freelyundergo axial deformation Rubber round gaskets are addedbetween each flange disc to play a sealing role

62 Establishment of Finite ElementModel of Inner Core SteelPipe e establishment of the correct finite element modelof the inner core steel tube is a necessary prerequisite for theestablishment of the finite element model of the compositelocal buckling fiber hinged rod erefore in this sectionfive core steel pipe finite element models with slendernessratios are established based on ANSYS softwaree elementadopts three-dimensional finite strain beam elementBEAM188 (suitable for analyzing various beam-columnmembers with large to medium slenderness ratios) and eachmember is divided into 20 elements (verified the number ofdivided elements meets the accuracy requirements) emodel created at this time has no geometric initial defectsand residual stress

e displacement constraint is shown in Figure 4 In themodel the y-axis passes through the center of the sectionalong the length of themember Since the circular section is asection with equal moment of inertia the x-axis is artificiallyassumed to be the weak axis so that the member undergoesfirst-order buckling around the x-axis erefore the tor-sional deformation and x-direction translational displace-ment of all nodes in the model are restricted (otherwisealthough static analysis can be carried out abnormalbuckling modes will occur) the z-direction displacement atboth ends of themember and the y-direction displacement atthe bottom end are also restricted

63 Finite Element Model of Hoop-Type Composite LocallyBuckled Fiber Hinged Compression Rod In this paper basedon the construction method of the flange-type compositelocal buckling fiber hinged rod the solid element (SOLID95)


finite element model and the beam element (BEAM188)finite element model are established to analyze the eigen-value buckling Among them the flange connection in thebeam element model adopts the simplified processing ofnode coupling

Both the core steel tube and the outer tube are simulatedby three-dimensional finite strain beam elements BEAM188e number of elements of the outer tube and the inner tubedepends on the length of different components rangingfrom 20 to 35 (to ensure that each finite element model has arelatively high accuracy)

e constitutive model of steel material in this paperadopts a multilinear follow-up strengthening model ecalculation process of finite element analysis is carried out inthree steps First we create a finite element model of aferrule-type composite locally buckled fiber hinged com-pression rod without initial defects and then perform elasticbuckling analysis e external load is continuously in-creased through cyclic iterations and finally the bucklingload factor is 10 thus obtaining the corresponding elasticbuckling load and first-order instability mode Finally theelastoplastic stability bearing capacity analysis of the localbuckling fiber hinged compression rod model of the high-strength steel composite is carried out Based on the uniformdefect model (that is the first-order modal solution based onthe eigenvalue buckling analysis) a peak value of L1000 isapplied en the arc length method is used to solve theproblem and the minimum number of substeps required forthe best effective solution is determined through continuousattempts to ensure the calculation accuracy and achieve thehighest calculation efficiency

7 High-Strength Steel Nonlinear DetectionResults and Analysis

71 Data Collection and Processing Properly we process theexperimental signals obtained through the nonlinear ul-trasonic system to obtain the magnitude of the fundamentalwave and the second harmonic and then calculate the valueof the ultrasonic nonlinear parameter Ultrasound isexpressed in two ways time domain signal and frequencydomain signal e time domain signal represents the re-lationship between the magnitude of the ultrasonic

amplitude and time e frequency domain signal reflectsthe amplitude distribution of the ultrasonic wave on thefrequency components

e pulse signal is excited by the RAM-5000 high-energyultrasound system passes through an attenuator for 9 dbattenuation enters a 04MHz low-pass filter for filteringand then is transmitted to a 12MHz ultrasonic piezoelectrictransducer e device converts it into an ultrasonic signaland transmits it to the samplee probe at the receiving endconverts the received ultrasonic signal into an electricalsignal e receiving probe transmits the signal to 16MHzhigh-pass filtering and 20 db preamplifier for processing andthen to the oscilloscope and computer for data processingand analysis What the computer extracts is the time domainsignal as shown in Figure 5 e signal collected by thecomputer can be processed by short-time Fourier transform(STFT) or fast Fourier transform (FFT) to obtain frequencydomain signals For the signal obtained in this experimentbecause the signals of different modes of the received signaloverlap each other in the time domain the effect of fastFourier transform is not gooderefore in this experimentthe short-time Fourier transform is used to process all thereceived signals

72 Research on the Relationship between Nonlinear Pa-rameters and Propagation Distance e FV520 B high-strength steel original plate specimen that has not undergonethe fatigue test is used as the experimental object to study therelationship between the ultrasonic nonlinear parametersand the propagation distance of the ultrasonic wave in theFV520B thin metal plate We keep the incident voltageunchanged and change the distance between the transmitterprobe and the receiver probe e distance between thetransmitter probe and the receiver probe is set to 40mm to80mm and a test is performed every 10mm increase

e relationship between the ultrasonic nonlinear pa-rameter β and the distance that the ultrasonic wave prop-agates in the FV520B sample is shown in Figure 6 At thistime the tested sample is the original FV520B high-strengthsteel sample without fatigue test e test results show thatthe nonlinear response can still be detected in the un-damaged intact sample indicating that the nonlinear re-sponse here is due to the inherent nature of the medium It

Load P

Load P

Load P

A

B

C

Aprime

Cprime

Bprime

0

Local buckling of tubeunder axial compression

Unstablebifurcation

buckling

Pcr

Pu

Figure 1 Local buckling and bifurcation instability of circular tube


can be considered that the internal microscopic features ofthe original sample are uniformly distributed and the ul-trasonic nonlinear response is caused by the nonharmonicityof the crystal lattice e experimental results show that theultrasonic nonlinear parameters increase approximatelylinearly as the propagation distance increases e resultsshow that as its propagation distance increases the ultra-sonic nonlinear response has a cumulative effect

e relationship between the ultrasonic nonlinear pa-rameter β and the propagation distance of the ultrasonicwave in the FV520B sample is shown in Figure 7 It can beconcluded from Figure 7 that as the propagation distanceincreases the ultrasonic nonlinear parameters have a cu-mulative effect and the results of the original test on theundamaged plate shape show that the nondamaged FV520Bhigh-strength steel medium itself has nonlinearity

According to the test results of notched specimens it can beseen that the contribution of cracks to the ultrasonic non-linear parameters is greater than the contribution of dis-locations to the ultrasonic nonlinear parameters e testedsample is a FV520B high-strength steel notched samplesubjected to a certain fatigue cycle After the fatigue testcracks appear in the notch of the notched sample Becausethe specimen is in a state of stress concentration at the notchthe stress it receives is much higher than the area far awayfrom the notch After the fatigue test a large number ofcracks appear near the notch and no cracks are found in thearea far away from the notch which stays before the crackinitiation e dislocation stage is shown in Figure 8 is isconsistent with the idea that the ultrasonic nonlinear re-sponse induced by cracks is greater than the ultrasonicnonlinear response induced by dislocations

Run script

Start

Add keywords

Trigger Bat fileBuild a finite element

model for elasticbuckling (Buckle)

Batch modifyinp file

Output first-orderbuckling mode

displacement value

Build an elastic-plasticbuckling finite elementmodel (elastoplastic)

Use Python to modifyits inp files in

batches

Add the keywordimperfection in thespecified location

Introduce geometric initialdefects for each

elastoplastic buckling modelby uniform defect modulus

method

Use Dos command to submitelastic and elastoplastic

finite element calculationfiles in turn

Start to automaticallycalculate run script

Obtain elastoplasticcalculation results

(0db) file Elastoplastic calculationresults

Figure 2 Parametric elastoplastic buckling analysis process

Load PA

B

(a)

Load PA

B

(b)

Load PA

B

(c)

Load PA

B

(d)

Figure 3 Schematic diagram of inner tube buckling


73 Research on the Relative Position of Nonlinear ParametersandCracks betweenTwoSensors In this section the notchedsample after fatigue damage appears as the experimentalobject and the influence of the relative position of the crackbetween the two ultrasonic transducers on the ultrasonicnonlinear parameters is studied e distance between thetransmitting probe and the receiving probe is alwaysmaintained at 60mm and the distance between the trans-mitting probe and the crack center at the notch is 10mm20mm 30mm 40mm and 50mm Each time a group istested the transmitting probe and the receiving probe mustmove 10mm at the same time to achieve the change of therelative position of the crack between the two probes

Specimen No 03 is a test piece of group A notch test(550MPa) cycled for 105 cycles e relationship betweenthe ultrasonic nonlinear parameters of No 03 specimen andthe distance between the transmitting probe and the crack isshown in Figure 9 e test results show that the size of theultrasonic nonlinear parameters will change with the po-sition of the crack between the two sensors

Specimen No 04 is a specimen of group B notch ex-periment (550MPa) cycled for 105 cycles e relationshipbetween the ultrasonic nonlinear parameters of the speci-men and the distance between the transmitting probe andthe crack is shown in Figure 10 It can be seen from the figurethat unlike the test result of No 03 specimen the magnitudeof the ultrasonic nonlinear parameter fluctuates greatly withthe change of the position of the crack at the two endsbetween the two sensors

Comparing the nonlinear ultrasonic detection results ofthe above two experiments it can be concluded that thechange of the position of the crack between the two sensorshas a greater impact on the ultrasonic nonlinear parametersComparing Figure 9 and Figure 10 it can be found that theultrasonic nonlinear parameter of No 04 sample in group Bis larger than the value of the ultrasonic nonlinear parameterof No 03 sample in group Ais is because the fatigue cyclecycles of these two specimens are both 105 times and theloading stress (660MPa) of the No 04 specimen in group Bis greater than the loading stress of the No 03 specimen ingroup A (550MPa) is shows that the greater the loadingstress of the fatigue experiment the greater the value of thecorresponding ultrasonic nonlinear parameter ere is amapping relationship between the nonlinear coefficient andthe degree of fatigue damage of FV520B high-strength steel

74 Research on the Variation Law of Nonlinear Parameterswith Fatigue Cycles In order to study the change law ofultrasonic nonlinear parameters of fatigue specimens indifferent cycles of fatigue cycles a group of plate-shapedspecimens and two groups of fatigue specimens with dif-ferent loading stresses are used as the research objects andthe different cycle cycles of each group are analyzed esecond fatigue specimen is tested e relationship betweennormalized nonlinear parameters and fatigue cycle is used todescribe the nonlinear changes of materials due to fatiguedamage

Model displacementconstraints

x

yz

High-strengthsteel finite

element modelUx = 0 Uy = 0 Uz = 0

ROTy = 0

Ux = 0

Figure 4 Schematic diagram of model displacement constraint

0 100 200 300 400 500 60004

08

12

16

2

24

Time (s)

Sign

al (M

Hz)

Figure 5 Time domain signal extracted by experiment


In order to study the effect of fatigue crack growth onultrasonic nonlinear parameters we designed two sets ofnotched specimens Figure 11 is the relationship curve betweennormalized ultrasonic nonlinear parameters and fatigue cyclecycles of the notched specimens in group A (550MPa) Fig-ure 12 shows the relationship between the normalized ultra-sonic nonlinear coefficients of the notched specimens in groupB (660MPa) and fatigue cycle cycles Ultrasonic nonlinearparameters are very sensitive to the fatigue damage of FV520Bhigh-strength steelmaterialse relationship between FV520Bhigh-strength steelrsquos nonlinear parameters and fatigue cyclecycles can be used to characterize its early fatigue

By studying the relationship between ultrasonic non-linear parameters and fatigue cycle cycles it can be foundthat there is a good correlation between the ultrasonicnonlinear parameters of FV520B high-strength steel andfatigue cycle cycles Nonlinear parameters can be used toreflect the fatigue life of materials In engineering practiceafter obtaining enough sample data points the nonlinearparameter curve of the engineering component of FV520Bhigh-strength steel material is calibrated in advanceComparing the nonlinear detection results with the cali-bration curve obtained in advance the fatigue life can beevaluated

10 20 30 40 50 60 70 80 90 10009

12

15

18

21

24

27

Propagation distance (mm)N

onlin

ear p

aram

eter

val

ue

Figure 6 e relationship between ultrasonic nonlinear parameter β and ultrasonic propagation distance

0 10 20 30 40 50 60 70 80 90 10012

15

18

21

24

27

Propagation distance (mm)

Non

linea

r par

amet

er v

alue

Figure 7 e relationship between the ultrasonic nonlinear coefficient β and the distance between the transducer and the gap

(a) (b)

Figure 8 Electron micrograph of the cross section of the notched sample (a) away from the notched area and (b) the notched area


0 10 20 30 40 50 60 70 80 90 1001819

2212223242526

Distance between launch probe and crack (mm)N

onlin

ear p

aram

eter

val

ue

Figure 9 e relationship between the ultrasonic nonlinear parameters of No 03 sample and the distance between the transmitting probeand the crack

0 10 20 30 40 50 60 70 80 90 100616263646566676869

Distance between launch probe and crack (mm)

Non

linea

r par

amet

er v

alue

Figure 10e relationship between the ultrasonic nonlinear parameters of No 04 sample and the distance between the transmitting probeand the crack

05 1 15 2 25 3 35 4 45 53

6

9

12

15

18

Fatigue cycle

Non

linea

r par

amet

er v

alue

times105

Figure 11e relationship between normalized ultrasonic nonlinear parameters and fatigue cycle cycles for notched specimens (550MPa)

3

4

5

6

7

8

9

Fatigue cycle

Non

linea

r par

amet

er v

alue

05 1 15 2 25 3 35 4 45 5times105



8 Conclusion

e outer steel tube combined with the local buckling fiberhinged compression rod has a good restraint effect on the innercore tube and the bearing capacity of the combined localbuckling fiber hinged compression rod increases with theincrease of the outer and inner stiffness ratio and the number ofhoops However an excessively large ratio of external andinternal stiffness will cause the instability mode of the com-posite locally buckled fiber hinged rod to change from commoninstability to the inner core alone which will reduce the bearingcapacity of the composite locally buckled fiber hinged rod Forthe core tube of high-strength and ultra-high-strength steel thestable bearing capacity of the composite local buckling fiberhinged compression rod becomes more obvious with the in-crease of the slenderness ratio of the core tubee influence ofrelated parameters in the dislocation string model and thedislocation couple model on the ultrasonic nonlinear pa-rameters is analyzed From the perspective of contact nonlinearacoustics the influence of cracks on ultrasonic nonlinearity isanalyzed and the finite element software ABAQUS is used tosimulate it It is proved that the crack can produce nonlineareffects and the relationship between nonlinear parameters andthe internal crack shape of thematerial is analyzede effect ofthe relative position of the crack between the two transducerson the ultrasonic nonlinear coefficient is studied and it can beconcluded that the change of the position of the crack betweenthe two transducers has little effect on the nonlinear coefficientwhich can be approximated as no effect By comparing twofatigue specimens with the same fatigue cycle but differentfatigue loading stress it is found that the larger the fatigue testloading stress the larger the corresponding ultrasonic non-linear parameter value ere is a mapping relationship be-tween the nonlinear coefficient and the degree of fatiguedamage Lamb waves are used to perform a series of nonlineartests on fatigue specimens of FV520B high-strength steelAnalyzing the results it is found that the material has a goodultrasonic nonlinear cumulative effect e experimental re-sults show that the ultrasonic nonlinear parameters have highsensitivity to the early fatigue damage of FV520B high-strengthsteel

Data Availability

Data sharing is not applicable to this article as no datasetswere generated or analyzed during the current study

Consent

Informed consent was obtained from all individual partic-ipants included in the study references

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

is work was supported by the Natural Science Foundationof the Jiangsu Higher Education Institutions of China (No

19KJB560018) and by the Jiangsu Province VocationalColleges Young Teacher Enterprise Practice Training Project(No 2020QYSJ165)

References

[1] D Wu Z Liu X Wang and L Su ldquoComposite magnetic fluxleakage detection method for pipelines using alternatingmagnetic field excitationrdquo NDT amp E International vol 91pp 148ndash155 2017

[2] J Abrahamsson M Hedlund T Kamf and H BernhoffldquoHigh-speed kinetic energy buffer optimization of compositeshell and magnetic bearingsrdquo IEEE Transactions on IndustrialElectronics vol 61 no 6 pp 3012ndash3021 2014

[3] P J Janse Van Rensburg A A Groenwold and D W WoodldquoOptimization of cylindrical composite flywheel rotors forenergy storagerdquo Structural and Multidisciplinary Optimiza-tion vol 47 no 1 pp 135ndash147 2013

[4] K M Lee S Y Park M Y Huh J S Kim and O EnglerldquoEffect of texture and grain size on magnetic flux density andcore loss in non-oriented electrical steel containing 315 SirdquoJournal of Magnetism and Magnetic Materials vol 354pp 324ndash332 2014

[5] P B Reddy A M El-Refaie S Galioto and J P AlexanderldquoDesign of synchronous reluctancemotor utilizing dual-phasematerial for traction applicationsrdquo IEEE Transactions onIndustry Applications vol 53 no 3 pp 1948ndash1957 2017

[6] Y Zhong L-E Rannar L Liu et al ldquoAdditive manufacturingof 316L stainless steel by electron beam melting for nuclearfusion applicationsrdquo Journal of Nuclear Materials vol 486pp 234ndash245 2017

[7] Y Gao G Y Tian K Li J Ji P Wang and H WangldquoMultiple cracks detection and visualization using magneticflux leakage and eddy current pulsed thermographyrdquo Sensorsand Actuators A Physical vol 234 pp 269ndash281 2015

[8] D S Petrovic B Markoli and M Ceh ldquoe nanostructure ofnon-oriented electrical steel sheetsrdquo Journal of Magnetismand Magnetic Materials vol 322 no 20 pp 3041ndash3048 2010

[9] S Ghanei M Kashefi and M Mazinani ldquoComparative studyof eddy current and Barkhausen noise nondestructive testingmethods in microstructural examination of ferrite-martensitedual-phase steelrdquo Journal of Magnetism and Magnetic Ma-terials vol 356 pp 103ndash110 2014

[10] I Tanaka H Nitomi K Imanishi K Okamura andH Yashiki ldquoApplication of high-strength nonoriented elec-trical steel to interior permanent magnet synchronous mo-torrdquo IEEE Transactions on Magnetics vol 49 no 6pp 2997ndash3001 2013

[11] S Sahin and M Ubeyli ldquoA review on the potential use ofaustenitic stainless steels in nuclear fusion reactorsrdquo Journalof Fusion Energy vol 27 no 4 pp 271ndash277 2008

[12] G Yang Z Zeng Y Deng et al ldquo3D EC-GMR sensor systemfor detection of subsurface defects at steel fastener sitesrdquoNDTamp E International vol 50 pp 20ndash28 2012

[13] Y Chang J Jiao G Li X Liu C He and B Wu ldquoEffects ofexcitation system on the performance of magnetic-flux-leakage-type non-destructive testingrdquo Sensors and ActuatorsA Physical vol 268 pp 201ndash212 2017

[14] J Gong and H W Luo ldquoProgress on the research of high-strength non-oriented silicon steel sheets in tractionmotors ofhybridelectrical vehiclesrdquo Journal of Materials Engineeringand Performance vol 43 no 6 pp 102ndash112 2015

[15] S Liu ldquoA new signal processing method based on notchfiltering and wavelet denoising in wire rope inspectionrdquo


Journal of Nondestructive Evaluation vol 38 no 2 pp 1ndash142019

[16] P Libeyre N Mitchell D Bessette Y Gribov C Jong andC Lyraud ldquoDetailed design of the ITER central solenoidrdquoFusion Engineering and Design vol 84 no 7ndash11 pp 1188ndash1191 2009

[17] P Zheng and J Zhang ldquoQuantitative nondestructive testing ofwire rope based on pseudo-color image enhancement tech-nologyrdquo Nondestructive Testing and Evaluation vol 34 no 3pp 221ndash242 2019

[18] Z Lv P Cai T Yu et al ldquoFatigue behaviors and damagemechanism of a Cr-Mn-N austenitic steelrdquo Journal of Alloysand Compounds vol 691 pp 103ndash109 2017

[19] M Zhao D Zhang and Z Zhou ldquoe research on quanti-tative inspection technology to wire rope defect based on Hallsensor arrayrdquo Nondestructive Testing vol 34 no 11pp 57ndash60 2012

[20] D L Zhang Y N Cao C Wang and D G Xu ldquoA newmethod of defects identification for wire rope based on three-dimensional magnetic flux leakagerdquo Journal of PhysicsConference Series vol 48 no 1 pp 334ndash338 2006

[21] H Ozeki K Hamada Y Takahashi et al ldquoEstablishment ofproduction process of JK2LB jacket section for ITER CSrdquoIEEE Transactions on Applied Superconductivity vol 24 no 3pp 1ndash4 2014

[22] E Arias-Castro and D L Donoho ldquoDoes median filteringtruly preserve edges better than linear filteringrdquo Annals ofStatistics vol 37 no 3 pp 1172ndash1206 2009

[23] A Nyilas ldquoFatigue crack growth rate and fracture toughnessof ITER central solenoid jacket materials at 7 Krdquo AIP Con-ference Proceedings vol 1435 no 1 pp 47ndash54 2012


specifications for high-strength steels simply apply designformulas suitable for ordinary-strength steels which greatlyrestricts the promotion and application of high-strengthsteel structures [9]

is paper proposes a simplified calculation model of thecomposite local buckling fiber hinged compression rod(using the BEAM188 element node coupling method) andthe correctness of the simplified calculation model is verifiedby comparison and analysis with solid elements In the initialdefect the residual stress will have an unfavorable effect onthe stable bearing capacity In this paper the analysis of thelocal buckling fiber hinged compression rod of high-strengthsteel with different slenderness ratios shows that the weldingresidual stress has little effect on the stable bearing capacity(when the residual stress is not considered the maximumdifference of the stable bearing capacity is only 302) Inthis paper the fabricated FV520B high-strength steel fatiguespecimen was tested by a nonlinear ultrasonic testing systemrough the data collection and processing the fundamentalwave and the second harmonic amplitude were extractede relationship between the ultrasonic nonlinear param-eters and the ultrasonic propagation distance in the materialwas studied e ultrasonic nonlinear response was detectedin the original plate specimen without fatigue damage in-dicating the inherent nonlinear characteristics of the ma-terial itself As the propagation distance increases theultrasonic nonlinear coefficient increases approximatelylinearly indicating that the nonlinear response has a cu-mulative effect Testing the notched samples that cycle for acertain number of cycles also found the cumulative effect ofthe ultrasonic nonlinear response e β-N curves ofFV520B material under three stress levels are obtained eresults show that the nonlinear parameters are very sensitiveto the early fatigue damage of FV520B high-strength steele nonlinear parameters show an increasing trend with theincrease of the number of fatigue cycles e relationshipbetween the ultrasonic nonlinear parameters of FV520Bhigh-strength steel and the fatigue cycle cycles can be used tocharacterize the early fatigue damage degree of the materialand reflect its fatigue life

2 Related Work

At present there are still few domestic studies on the overallstability and local stability of high-strength steel beams [10]Relevant scholars have studied the stability of high-strengthsteel thin-walled box-section beams mainly analyzing theeffects of the slenderness ratio flange width-to-thicknessratio and section side-length ratio of the box beam on itsoverall stability [11] e steel studied is 6mm thick18Mn2Cr Mo BA high-strength steel with a yield strength of745MPa is type of steel is generally used in bridge en-gineering Relevant scholars pointed out that the slendernessratio has little effect on the ultimate load of a box-shapedbeam in a purely bending state [12] After changing theslenderness ratio of the beam the ultimate load changeswithin 5 e flange width of the beam is relatively smallAfter the load-displacement curve appears at the highestpoint the curve slowly and steadily declines when the flange

width-to-thickness ratio of the beam is increased the flangeor web has already experienced local instability after the loadis added to the ultimate load [13] e elastic sectionmodulus of the beam decreases so the slope of thedescending section after the highest point of the load-dis-placement curve increases significantly It can be seen thatthe width-to-thickness ratio of the flange affects the overallstability of the beam to a greater extent When the cross-sectional side length ratio of the beam increases from 30 to60 the web height-to-thickness ratio continues to increaseand local instability is prone to occur in the compressionzone of the web which leads to an acceleration of the overallstability of the ultimate load drop [14]

e research on the local-total related buckling of box-shaped cross section members has a long history Domesticand foreign researches mainly focus on the local-total relatedbuckling of ordinary steel members e research on thelocal-total related buckling of high-strength steel membersmainly focuses on the axis Relevant scholars have usednumerical analysis methods to study the relative bucklingultimate bearing capacity of ordinary steel members withthin-walled box-shaped sections under axial compressionwith hinged ends [15] Research has found that when theultimate bearing capacity of the component is reached thebearing capacity of the component composed of weak panelswill decrease sharply [16] e researchers used the testresults of four axial compression members with a nominalyield strength of 390MPa to compare with the finite elementmodel [17] e research shows that residual stress and localoverall geometric defects have a great influence on the ul-timate bearing capacity of members Researchers haveproposed a formula for calculating the local-to-whole rel-ative buckling ultimate bearing capacity of axial compres-sion members with large width-to-thickness ratio based onthe effective width method [18] Relevant scholars haveconducted experimental research on three S355 steel weldedthin-walled box-shaped eccentric components established afinite element model considering the effects of initial geo-metric defects and residual stress and compared the ex-perimental results and they found that the two are in goodagreement Finally the verified finite element model is usedto study the influence of initial defects on the ultimatebearing capacity of the component [19]

Relevant scholars have respectively measured and ana-lyzed the overall geometric initial defects of I-shaped sectionmembers with nominal yield strengths of 460MPa and960MPa [20]e results show that themeasured amplitude ofthe overall geometric initial defects of most high-strength steelsis less than 11000e researchers measured and analyzed theoverall geometric initial defects of the I-shaped sectionmembers with nominal yield strengths of 460MPa 550MPaand 690MPa and the results showed that the maximumamplitudes were 1541 1339 and 1702 [21] Relevant scholarshave studied the effect of geometric initial defects on thestability of ordinary steel and high-strength steel [22] eresults show that compared with ordinary steel membersgeometric initial defects have less effect on the stability coef-ficient of high-strength steel members compared with theultimate bearing capacity the local buckling capacity is more













wGnabla8 +tEz

4w

rzx4 + Nxnabla

4z2w

zx2 0 (1)



w w1 sinπlx

x middot cosπly

y (2)


Nx Gπ2ψ minustE

r2π2ψ

(3)

Among them

ψ 1l2x


⎛⎝ ⎞⎠

2

(4)


Nx t middot E

r middot3 v

2minus 11113872 1113873

1113969 (5)


Nx 0582 middottE

r (6)





























Load P

Load P

Load P

A

B

C

Aprime

Cprime

Bprime

0


Unstablebifurcation

buckling

Pcr

Pu






Run script

Start

Add keywords





displacement value



batches




method







Load PA

B

(a)

Load PA

B

(b)

Load PA

B

(c)

Load PA

B

(d)









x

yz



ROTy = 0

Ux = 0


0 100 200 300 400 500 60004

08

12

16

2

24

Time (s)

Sign

al (M

Hz)





10 20 30 40 50 60 70 80 90 10009

12

15

18

21

24

27


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 10012

15

18

21

24

27


Non

linea

r par

amet

er v

alue


(a) (b)



0 10 20 30 40 50 60 70 80 90 1001819

2212223242526


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 100616263646566676869


Non

linea

r par

amet

er v

alue


05 1 15 2 25 3 35 4 45 53

6

9

12

15

18

Fatigue cycle

Non

linea

r par

amet

er v

alue

times105


3

4

5

6

7

8

9

Fatigue cycle

Non

linea

r par

amet

er v

alue

05 1 15 2 25 3 35 4 45 5times105



8 Conclusion


Data Availability


Consent




Acknowledgments



References






































wGnabla8 +tEz

4w

rzx4 + Nxnabla

4z2w

zx2 0 (1)



w w1 sinπlx

x middot cosπly

y (2)


Nx Gπ2ψ minustE

r2π2ψ

(3)

Among them

ψ 1l2x


⎛⎝ ⎞⎠

2

(4)


Nx t middot E

r middot3 v

2minus 11113872 1113873

1113969 (5)


Nx 0582 middottE

r (6)





























Load P

Load P

Load P

A

B

C

Aprime

Cprime

Bprime

0


Unstablebifurcation

buckling

Pcr

Pu






Run script

Start

Add keywords





displacement value



batches




method







Load PA

B

(a)

Load PA

B

(b)

Load PA

B

(c)

Load PA

B

(d)









x

yz



ROTy = 0

Ux = 0


0 100 200 300 400 500 60004

08

12

16

2

24

Time (s)

Sign

al (M

Hz)





10 20 30 40 50 60 70 80 90 10009

12

15

18

21

24

27


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 10012

15

18

21

24

27


Non

linea

r par

amet

er v

alue


(a) (b)



0 10 20 30 40 50 60 70 80 90 1001819

2212223242526


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 100616263646566676869


Non

linea

r par

amet

er v

alue


05 1 15 2 25 3 35 4 45 53

6

9

12

15

18

Fatigue cycle

Non

linea

r par

amet

er v

alue

times105


3

4

5

6

7

8

9

Fatigue cycle

Non

linea

r par

amet

er v

alue

05 1 15 2 25 3 35 4 45 5times105



8 Conclusion


Data Availability


Consent




Acknowledgments



References















































Load P

Load P

Load P

A

B

C

Aprime

Cprime

Bprime

0


Unstablebifurcation

buckling

Pcr

Pu






Run script

Start

Add keywords





displacement value



batches




method







Load PA

B

(a)

Load PA

B

(b)

Load PA

B

(c)

Load PA

B

(d)









x

yz



ROTy = 0

Ux = 0


0 100 200 300 400 500 60004

08

12

16

2

24

Time (s)

Sign

al (M

Hz)





10 20 30 40 50 60 70 80 90 10009

12

15

18

21

24

27


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 10012

15

18

21

24

27


Non

linea

r par

amet

er v

alue


(a) (b)



0 10 20 30 40 50 60 70 80 90 1001819

2212223242526


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 100616263646566676869


Non

linea

r par

amet

er v

alue


05 1 15 2 25 3 35 4 45 53

6

9

12

15

18

Fatigue cycle

Non

linea

r par

amet

er v

alue

times105


3

4

5

6

7

8

9

Fatigue cycle

Non

linea

r par

amet

er v

alue

05 1 15 2 25 3 35 4 45 5times105



8 Conclusion


Data Availability


Consent




Acknowledgments



References






























Run script

Start

Add keywords





displacement value



batches




method







Load PA

B

(a)

Load PA

B

(b)

Load PA

B

(c)

Load PA

B

(d)









x

yz



ROTy = 0

Ux = 0


0 100 200 300 400 500 60004

08

12

16

2

24

Time (s)

Sign

al (M

Hz)





10 20 30 40 50 60 70 80 90 10009

12

15

18

21

24

27


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 10012

15

18

21

24

27


Non

linea

r par

amet

er v

alue


(a) (b)



0 10 20 30 40 50 60 70 80 90 1001819

2212223242526


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 100616263646566676869


Non

linea

r par

amet

er v

alue


05 1 15 2 25 3 35 4 45 53

6

9

12

15

18

Fatigue cycle

Non

linea

r par

amet

er v

alue

times105


3

4

5

6

7

8

9

Fatigue cycle

Non

linea

r par

amet

er v

alue

05 1 15 2 25 3 35 4 45 5times105



8 Conclusion


Data Availability


Consent




Acknowledgments



References

































x

yz



ROTy = 0

Ux = 0


0 100 200 300 400 500 60004

08

12

16

2

24

Time (s)

Sign

al (M

Hz)





10 20 30 40 50 60 70 80 90 10009

12

15

18

21

24

27


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 10012

15

18

21

24

27


Non

linea

r par

amet

er v

alue


(a) (b)



0 10 20 30 40 50 60 70 80 90 1001819

2212223242526


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 100616263646566676869


Non

linea

r par

amet

er v

alue


05 1 15 2 25 3 35 4 45 53

6

9

12

15

18

Fatigue cycle

Non

linea

r par

amet

er v

alue

times105


3

4

5

6

7

8

9

Fatigue cycle

Non

linea

r par

amet

er v

alue

05 1 15 2 25 3 35 4 45 5times105



8 Conclusion


Data Availability


Consent




Acknowledgments



References





























10 20 30 40 50 60 70 80 90 10009

12

15

18

21

24

27


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 10012

15

18

21

24

27


Non

linea

r par

amet

er v

alue


(a) (b)



0 10 20 30 40 50 60 70 80 90 1001819

2212223242526


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 100616263646566676869


Non

linea

r par

amet

er v

alue


05 1 15 2 25 3 35 4 45 53

6

9

12

15

18

Fatigue cycle

Non

linea

r par

amet

er v

alue

times105


3

4

5

6

7

8

9

Fatigue cycle

Non

linea

r par

amet

er v

alue

05 1 15 2 25 3 35 4 45 5times105



8 Conclusion


Data Availability


Consent




Acknowledgments



References



























0 10 20 30 40 50 60 70 80 90 1001819

2212223242526


onlin

ear p

aram

eter

val

ue


0 10 20 30 40 50 60 70 80 90 100616263646566676869


Non

linea

r par

amet

er v

alue


05 1 15 2 25 3 35 4 45 53

6

9

12

15

18

Fatigue cycle

Non

linea

r par

amet

er v

alue

times105


3

4

5

6

7

8

9

Fatigue cycle

Non

linea

r par

amet

er v

alue

05 1 15 2 25 3 35 4 45 5times105



8 Conclusion


Data Availability


Consent




Acknowledgments



References



























8 Conclusion


Data Availability


Consent




Acknowledgments



References



























nonlinear analysis method of high-strength steel based on

Documents