duan fangjuan liu jingxi xie de

Received：2018 - 03 - 30Supported by: National Natural Science Foundation of China (51579110)Author(s): DUAN Fangjuan, female, born in 1993, master degree candidate. Research interests：structural collision, structure de⁃

fect and impact mechanical property. E-mail: [email protected] Jingxi, male, born in 1975, Ph.D., associate professor. Research interests：structural collision, fatigue and fracture,ship structure design and analysis. E-mail: [email protected] De, male, born in 1964, Ph.D., professor. Research interests：structural collision, fatigue and fracture, design andmanufacture of ships and marine structures. E-mail: [email protected]

*Corresponding author：LIU Jingxi

Dynamic response of aluminum alloyplate with initial cracksunder repeated impacts

Duan Fangjuan1，2，Liu Jingxi*1，2，Xie De1，2

1 School of Naval Architecture and Ocean Engineering，Huazhong University of Science and Technology，Wuhan 430074，China

2 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration，Shanghai 200240，ChinaAbstract：［Objectives］This paper aims to study the effects of initial cracks on the dynamic response of aluminumalloy plate under repeated impacts.［Methods］Through the repeated impact test and finite element simulation of thealuminum alloy plate，we analyze the dynamic response characteristics of the aluminum alloy plate under repeatedimpacts，and compare the impact loads and failure modes of the aluminum alloy plate with and without initial cracks.［Results］The test results are in good agreement with the simulation calculations. The results show that the aluminumalloy plate is sensitive to initial cracks under repeated impacts；the initial cracks can reduce the load carrying capacityof the aluminum alloy plate，resulting in the reduced impact load and the reduced number of the repeated impacts forfailure；the failure modes are also influenced by the initial cracks under repeated impacts.［Conclusions］ Theresearch in this paper provides a certain basis and reference for calculating and evaluating the structure strength ofaluminum alloy hull plate.Key words：repeated impacts；load carrying capacity；failure modes；aluminum alloy plate with initial cracks；impacttest；numerical simulationCLC number: U661.43

To cite this article：Duan F J, Liu J X, Xie D. Dynamic response of aluminum alloy plate with initial cracks under repeated impacts[J/OL]. Chinese Journal of Ship Research, 2019, 14(2). http://www.ship-research.com/EN/Y2019/V14/I2/63.

DOI：10.19693/j.issn.1673-3185. 01231

0 Introduction

Ships and marine engineering structures are oftensubject to repeated impacts, such as the impact ofwaves on bows and sterns, the impact of ice loads,and collisions between ships and docks when shipsdocked. Under repeated impacts, when initial cracksexist in the ship shell plate, the load carrying capaci⁃ty of the hull will decrease sharply, leading to largedeformation even failure and destruction of the hull.At present, there is no design specification clearly in⁃dicate the ultimate bearing state of the outer panel ofthe ship with initial defects under repeated impacts.Therefore, the research and analysis of the ultimate

load carrying capacity is particularly important forship safety. To this end, some scholars have studiedthe mechanical properties of different structures un⁃der the action of repeated impacts. Huang et al. [1]

carried out an experiment on a set of rigid sphericalpunches repeatedly striking a solid-supported alumi⁃num alloy round plate on a drop-hammer impactloading device. They found that for elastic-plasticstructures with boundary constraints, when the struc⁃ture was subjected to the same repeated dynamicload and elasto-plastic deformation, the stored elas⁃tic deformation energy will increase with the rise ofdeformation. Rajkumar et al. [2] carried out four re⁃peated impact tests on the aluminum alloy plate, and

CHINESE JOURNAL OF SHIP RESEARCH，VOL.14，NO.2，APR 2019 23

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CHINESE JOURNAL OF SHIP RESEARCH，VOL.14，NO.2，APR 2019carried out a low-speed drop-hammer test on thefour-edge clamp supported specimen, with a hemi⁃spherical punch of 5.2 kg. Yue et al. [3] used thedrop-hammer impact tester to carry out the repeatedimpact test of composite plates, and compared the en⁃ergy absorption characteristics and failure modes ofcomposite plates under different impact energy.

For ships and offshore platforms, crack-like de⁃fects are inevitable in their structure. These defectsmay be inherent in the material or may be caused bymanufacturing processes or the impact during opera⁃tion, which should be taken into account in the engi⁃neering analysis of ships and marine structures. Inrecent years, the influence of initial defects on the re⁃sidual load carrying capacity and the failure modehas attracted more and more attention. Zhang et al. [4]

numerically simulated the process of cracked platestretching using the element removal method. Fan etal. [5] tested the center cracked plates with differentwidths and different initial crack lengths to investi⁃gate the geometric factors affecting the fracturetoughness and residual strength of the crack-contain⁃ing aluminum alloy sheet material, finding that boththe specimen width and the initial crack length hadan effect on fractured toughness. Yin et al. [6] used thetheoretical method to study the dynamic response ofthe structure with initial cracks, established the dy⁃namic reliability analysis and life assessment modelof the structure with reduced strength, and discussedthe impact of the initial crack change on reliability.Based on numerical calculations and experimentalresults, Paik et al. [7-9] derived the ultimate strengthcalculation method for the plate with initial cracks,finding that the transverse crack whose length direc⁃tion is perpendicular to the loading direction wouldgreatly reduce the ultimate load carrying capacity ofthe plate, while the longitudinal crack whose lengthdirection was parallel to the loading direction had asignificantly smaller reduction effect in load carryingcapacity than the transverse crack. Zhang et al. [10]

used the nonlinear finite element method to studythe ultimate strength of stiffened plates with cracksand pitting damage under axial compression, findingthat the increase of crack length and pitting damagewould significantly reduce the residual strength ofthe stiffened plates. Seifi et al. [11] studied the ulti⁃mate load carrying capacity of aluminum alloy platewith initial cracks under static load. In addition,some scholars have carried out research on the influ⁃ence of the initial crack length, position and anglewith the loading direction on the load carrying capac⁃

ity of the light plate and the stiffened plate [12-13].According to the analysis of the above references,

the current research on load carrying capacity forsimple plate structures mostly focuses on consider⁃ing a single influencing factor, namely only crack oronly the number of impacts, and rarely the couplingeffect of the two. In fact, under repeated impacts, theexistence of cracks will change the failure mode ofthe plate, thus affecting the plate's bearing perfor⁃mance. Analysis of the two together is of great signifi⁃cance for the evaluation of the ultimate load carryingcapacity of the ship shell plates. This paper intendsto carry out the repeated impact test of aluminum al⁃loy plates with and without initial cracks on thedrop-hammer tester, so as to explore the change ofimpact force with impact time and impact times, com⁃pare the failure modes of the two, and analyze the im⁃pact of the initial crack. At the same time, a simula⁃tion model is established based on the test to com⁃pare the test and simulation results, so as to moredeeply analyze the failure mechanism of the alumi⁃num alloy plate under repeated impacts.1 Materials and experimental re-

search

1.1 Material model

The material used in the test model herein wasAA5083-H116. Although steel is a traditional mate⁃rial in ships and marine engineering structures, alu⁃minum alloy materials have been increasingly usedin the construction of high-speed craft since the1990s. AA5083-H116 is a high-magnesium alloywith excellent strength, which is also the non-heattreatable alloy material with endurance, corrosion re⁃sistance and machinability.

In the finite element simulation, when the materialwas in the elastic deformation stage, the true stresswas linear with the true strain, and its constitutiveequation was Hooke's law. When the strain exceededthe elastic limit and the material was in the elasto⁃plastic deformation stage, there was a nonlinear rela⁃tionship between the true stress σ and the truestrain ε , its constitutive equation could be ex⁃pressed as

σ = σs + B( )ε - εs

n（1）

where σs is the elastic limit stress; εs is the elasticlimit strain; B is the strengthening coefficient; n isthe strengthening index. In this test, referring toGB228-87 Tensile Test Method of Metal, the engi⁃

24


Server system

Fig.2 Schematic diagram of electro-spark-erosion crack prefabrication

Pulsepowersupply

Automatic feeding device

Tool electrode

SpecimenWorking fluid

Fixed device Filter Liquidpump

Working fluid

neering stress-strain curve of the material was ob⁃tained by the specimen tensile test, and the truestress–strain curve shown in Fig. 1 was obtained byfitting with Eq. (1). Table 1 gives the basic mechani⁃cal parameters of the material.

1.2 Specimen size

The aluminum alloy plates with and without initialcracks had a geometry of 100 mm × 100 mm × 6 mmand were cut off from a plate with a size of 1 000 mm× 1 000 mm × 6 mm. The length of the crack was8 mm, which is equal to the diameter of the punch.The crack depth was 1 mm, namely that the ratio ofcrack depth to plate thickness was 1/6. In the test,the square specimen was fixed by 12 equal-pitch

hexagon socket screws between the upper and lowersteel splints. The two splints had the same geometri⁃cal dimension of 100 mm × 100 mm × 10 mm, and acircular region with a diameter of 75 mm was cut offfrom the middle to serve as the impact test area.1.3 Crack prefabrication

The initial crack on the specimen was prefabricat⁃ed by the electric-spark machining process. The pro⁃cessing principle is shown in Fig. 2. Electric-sparkmachining, also called electric discharge machining,is a method of processing materials by means of elec⁃trical corrosion caused by pulse discharge betweentwo poles. When the tool electrode (positive elec⁃trode) and the workpiece (negative electrode) im⁃mersed in the liquid electrolyte are close in the insu⁃lator, the interelectrode voltage ionizes the "relative⁃ly closest point" between the two electrodes to form apulse discharge, which gradually corrodes the excessmetal.

The electrode of the electric-spark machining toolwas made of pure copper with a density of 8 933 kg/m3.It could be moved in one direction at a constantspeed in the vertical and horizontal directions withan accuracy of 0.03 mm. First, the tool electrode wasfed vertically downward along the Z axis, and the fi⁃nal position on the Z axis was determined by thedepth of the precrack. Then, the electrode was fedalong the X axis with a constant discharge gap be⁃tween the electrode and the specimen, and the feeddistance was determined by the length of the pre⁃crack. The crack prefabrication process is shown inFig. 3. The working fluid continuously circulated dur⁃ing the process, taking away the heat and corrosiondebris generated during the discharge. Fig. 4 showsthe specimen after prefabrication of a crack of 8 mmlong and 1 mm deep.

ParameterDensity ρ /（kg·m-3）

Young's modulus E/GPaPoisson's ratio μ

Elastic limit stress σs /MPaElastic limit strain εs

Strengthening coefficient B

Strengthening index n

Value700680.32380.0013850.068

Table 1 Mechanical properties of AA5083-H116 material

Fig. 1 True tensile stress-strain curve of the specimen

500450400350300250200 0 0.05 0.10 0.15 0.20 0.25

True strain ε

Truest

ressσ

/MPa

Duan F J, et al. Dynamic response of aluminum alloy plate with initial cracks under repeated impacts 25


CHINESE JOURNAL OF SHIP RESEARCH，VOL.14，NO.2，APR 2019

1.4 Test procedures

The repeated impact test was conducted on adrop-hammer tester, as shown in Fig. 5. The punchwas hemispherical with a diameter of 12 mm and amass of 13.26 kg. The impact energy used in this pa⁃per was 60 J. Through a lot of tests, it was found thatthe specimen can be pierced within the appropriatenumber of repeated impacts at this energy.

First, the specimen was bolted between the twosplints to prevent from sliding horizontally. Then, thefixed specimen and the two splints were placed to⁃gether on the test bench, and fixed by the clamping

device to limit the degree of freedom in the verticaldirection of the perimeter of the specimen. Aftereach impact, there would be a rebound catcher tocatch the punch to prevent secondary impact. For aspecimen with initial cracks, the face with crackswas facing down and not in direct contact with thepunch.

After each impact, the plastic deformation of thefront and back of the specimen was recorded by adigital camera, and the deformation of the specimenalong the centerline was measured by a laser dis⁃placement sensor. The measuring device of deforma⁃tion is shown in Fig. 6. The impact force between thepunch and the specimen was measured by the forcesensor. The impact test was repeated with the impactenergy of 60 J until the specimen was significantlydamaged. After the test was completed, theforce-time curve, the displacement-time curve, andthe energy-time curve were all output from the com⁃puter.

2 Numerical simulation

2.1 Finite element model

The numerical simulation of the repeated impacttest was performed using the nonlinear finite elementsoftware ABAQUS/Explicit. The aluminum alloyplates with and without initial cracks were numerical⁃ly modeled according to the experimental setup. For

（b）Partial enlargementFig.3 Machining process of crack on the specimen

SpecimenTool electrode

Feeding directionFeeding direction

（a）Specimen diagram

SpecimenTool electrode

Fig.4 Specimen after crack prefabrication

Specimen

Precrack8 mm

Fig.5 The impact testing machine

Impact mass block

Hemisphericalpunch

Drop-hammertrack

Clamping deviceFig.6 The deflection measurement set-up

Specimen ofaluminium plate

Laser displacement sensor

A laboratory bench movingdownward at a constant speed

26


the aluminum alloy plate with cracks, the initial de⁃fects were created by removing the mesh at the cen⁃ter of the plate to simulate real cracks. The finite ele⁃ment model is shown in Fig. 7. The solid elementC3D8R was used, and the punch was set as a rigidbody. The upper and lower splints fixed the upperand lower surfaces according to the boundary in thetest, and the punch only had the degree of freedomin the vertical direction. In order to improve the accu⁃racy of the calculation and efficiency, the mesharound the impact point was locally optimized.

2.2 Failure criterion

The aluminum alloy is an isotropic elastoplasticmaterial. The parameters are shown in Table 1, andthe stress-strain curve of the plastic phase is shownin Fig. 1 (regardless of the influence of temperature).The failure mode of the aluminum alloy plate was de⁃fined by the ductile material failure model ofABAQUS. The damage initiation and damage evolu⁃tion were controlled by the equivalent fracture strainωD and the fracture energy G f , respectively.

When the model unit satisfied Eq. (2), the failure

began.ωD = dε

pl

εpl

D ( )η εpl

= 1 （2）

where εpl , εpl

D , εpl

D and η represent equivalent plas⁃tic strain, ultimate equivalent plastic strain, equiva⁃lent plastic strain rate and stress triaxiality, respec⁃tively.

The damage extension after reaching the initialdamage was controlled by the fracture energy G f . Inother words, when the unit satisfied Equation (3), itwas judged to be failed and was deleted from themodel.

G f = 0

upl

f

σy upl

（3）where u

pl

f and σy represent equivalent plastic defor⁃mation and yield stress, respectively.3 Test results

3.1 Result of intact aluminum alloy plateunder repeated impacts

Fig. 8 shows the test and simulation curves of thecarrying capacity of the intact aluminum alloy platewith time in the impact test. It can be seen from thefigure that the test and simulation match well. Thesimulation results are slightly larger than the test re⁃sults, and the error is mainly caused by the inappro⁃priate cell size. In the first six repeated impacts, asthe number of impacts increases, the impact force isgradually increased due to the hardening of the mate⁃rial, and the impact time is successively shortened.At the sixth impact, the impact force reaches a peak.At this time, although the plate begins to break, itstill has a certain load carrying capacity. After thesixth impact, due to the accumulation of damage, anew crack is generated at the impact point, so thatthe load carrying capacity of the plate is rapidly re⁃

（a）Intact model

（b）Back of specimen model

Punch

Splint

Aluminum plate

crack

（c）Enlargement of initial crackFig.7 FE model of repeated impact simulation

Fig.8 The impact force-time curves of the intact specimen

35302520151050

Impact

force/k

N

0 10 20 30 40 50 60Time/ms

Test valueSimulation value



CHINESE JOURNAL OF SHIP RESEARCH，VOL.14，NO.2，APR 2019duced, and the impact force begins to decrease. Thespecimen breaks down until the 9th impact.

The plastic deformation profile of the plate aftereach impact in the impact test is shown in Fig. 9.The asymmetry of the last impact deformation curveis caused by the crack of the plate. The damage pat⁃terns of the specimen under the first, third, sixth andninth impacts are shown in Fig. 10. As can be seenfrom the figure, the deformation of the specimen canbe divided into two parts: the overall deformationand the local pit. The part that fits the punch is a par⁃tial pit, which is generated by the bending and shear⁃ing caused by the downward movement of the punch.

The deformation from the necking ring to the bound⁃ary segment is a global deformation, which is theplastic deformation caused by the membrane stretch⁃ing. The shape of the local pit is consistent with thatof the punch. As the number of impacts increases,the deformation of the specimen rises until it fails,and the ratio of the local depression value to the over⁃all deformation value increases successively. This in⁃dicates that in the process of repeated impacts, localpits play an important role in the response of theplate and should be paid with enough attention.

Fig. 9 Deformed shape profile of intact specimen after each impact

Distance from the impact point/mm-40 -30 -20 -10 0 10 20 30 40

Impact piercing

Plastic deformation/mm

0-2-4-6-8

-10

The 1st impactThe 2rd impactThe 3rd impactThe 4th impact

The 5th impactThe 6th impactThe 7th impactThe 8th impactThe 9th impact

（a）Front face of the plate

Front face

1st impact 10 mm

Front face

3rd impact10 mm

Front face

6th impact 10 mm

Front face

9th impact 10 mm

（b）Black face of the plateFig.10 The impact damages of the intact specimen

Back face

1st impact 10 mm

Back face

3rd impact 10 mm

Back face

6th impact 10 mm

Back face

9th impact 10 mm

3.2 Results of aluminum alloy plate withinitial cracks under repeated impacts

Fig. 11 shows the carrying capacity-time curve ofthe aluminum alloy plate with initial cracks in theimpact test. The test and simulation results agreewell, and the simulation results are slightly largerthan the test results. It can be seen from the figurethat the aluminum alloy plate with initial cracks failsafter repeated impacts for eight times under the im⁃pact energy of 60 J, which is one time less than thenumber of impacts on the intact aluminum alloyplate. As the number of impacts increases, the carry⁃ing capacity of the plate increases, reaching the max⁃imum at the 5th impact, and then gradually decreas⁃ing until it breaks. Fig. 12 shows a comparison of the

maximum carrying capacity of aluminum alloy plateswith and without initial cracks. It can be seen fromthe figure that the carrying capacity of the aluminumalloy plate with cracks in each impact is significantlysmaller than that of the intact aluminum alloy plate.Before reaching the peak, the carrying capacity isproportional to the number of impacts, which iscaused by the hardening of the material. It can beseen from the true stress-strain curve of the alumi⁃num alloy in Fig. 1 that the material has no obviousyielding stage, and the true stress increases withstrain, so the carrying capacity increases with thenumber of impacts. The number of impacts when thecarrying capacity of the aluminum alloy plate withcracks reaches the peak and the number of impactsat the time of breaking are both one time earlier than

28


those of intact aluminum alloy plate, indicating thatthe initial crack causes the rigidity of the plate to de⁃crease and has a great influence on the load carryingcapacity of the aluminum alloy plate.

Comparing Fig. 10 and Fig. 13, we can find thataluminum alloy plates with and without initial cracksexhibit different failure modes under repeated im⁃pacts. Under the repeated impacts, the intact alumi⁃num alloy plate first forms a pit at the impact point,and as the number of impacts increases, a crack isgenerated on the necking circle of the pit. The occur⁃rence of the crack causes the load carrying capacityof the plate to drop rapidly, and as the number of im⁃pacts increases, the plate is soon cracked. Under therepeated impacts, the aluminum alloy plate with ini⁃tial cracks expands along the length of the initial

crack to the sides. As the number of impacts increas⁃es, a crack is generated along the edge of the punchpit in the direction perpendicular to initial crack. Asa result, the plate cracks and penetrates in the direc⁃tion of crack initiation with the initial crack as theboundary.4 Conclusions

1) As the number of impacts increases, the impactforce of the aluminum alloy plate begins to increasedue to the hardening of the material. When the crackoccurs in the pit area, due to the sharp decrease inload carrying capacity, the aluminum alloy plate fi⁃nally breaks.

2) The appearance of the initial crack will serious⁃ly reduce the load carrying capacity of the aluminumalloy plate, so that the maximum number of impactsof aluminum alloy plate is reduced, and the carryingcapacity is also reduced. The nonlinear dynamic fi⁃nite element software can be used to describe the re⁃peated impacts of aluminum alloy plate with initialcracks, and the feasibility of this method has beenproved by experiments. This crack simulation meth⁃od is of great significance for evaluating the load car⁃rying capacity of the hull shell plate with initial de⁃fects, ship structure design and optimization direc⁃tion.

3) The aluminum alloy plates with and without ini⁃tial cracks have different failure modes under repeat⁃ed impacts. Under repeated impacts, the intact alu⁃minum alloy plate is broken along the pit aftercracks occur in the pit area. The aluminum alloyplate with the initial cracks produces transversecracks along the direction perpendicular to the cracklength direction, which is then torn along the crackdirection.References［1］ Huang Z Q，Chen Q S，Zhang W T. Pseudo-shake⁃

down in the collision mechanics of ships［J］. Interna⁃tional Journal of Impact Engineering，2000，24（1）：

19-31.［2］ Rajkumar G R，Krishna M，Murthy H N N，et al. In⁃

vestigation of repeated low velocity impact behaviour ofGFRP/aluminium and CFRP/aluminium laminates［J］.International Journal of Soft Computing & Engineer⁃ing，2012，6（1）：50-58.

［3］ Yue H L，Zhang G L，Wang Z Z，et al. Evaluation ofressistance to repeated low-velocity impact of compos⁃ite laminates with polyurethane coating［J］. ChineseJournal of Materials Research， 2016， 30 （5）：

379-387（in Chinese）.

Fig.11 The impact force-time curves of the specimen with aninitial crack

302520151050

Carryin

gcapa

city/kN

0 10 20 30 40 50Time/ms

Test valueTest valueSimulation valueSimulation value

343230282624222018

Maxim

umcar

ryingc

apacity

/kN

0 2 4 6 8 10Number of intacts

Impact forceNumber of intacts

Fig.12 The maximum impact force-impact number curves ofthe intact and cracked plates

（a）Test result （b）Simulation resultFig.13 The damage modes of the specimen with crack

10 mm 10 mm



CHINESE JOURNAL OF SHIP RESEARCH，VOL.14，NO.2，APR 2019

含初始裂纹铝合金板在反复冲击载荷下的动态响应

段芳娟 1，2，刘敬喜*1，2，解德 1，2

1 华中科技大学船舶与海洋工程学院，湖北武汉 4300742 高新船舶与深海开发装备协同创新中心，上海 200240

摘要：［目的目的］为研究含初始裂纹铝合金板在反复冲击载荷作用下的动态响应影响，［方法方法］通过开展铝合金

板的反复冲击试验和有限元仿真研究，分析在反复冲击载荷下铝合金板的动态响应特性，比较完整铝合金板和

含初始裂纹铝合金板在反复冲击载荷作用下的冲击力和破坏模式。［结果结果］试验结果与仿真计算取得较好的

吻合。结果表明，在反复冲击载荷作用下，铝合金板对初始裂纹较为敏感；初始裂纹会降低铝合金板的承载能

力，使得冲击力减小，反复冲击直至失效的冲击次数减小；含初始裂纹铝合金板的破坏模式也会受到影响。［结结

论论］研究结果可对铝合金船体外板的结构强度计算和评估提供一定的依据和参考。

关键词：反复冲击；承载能力；破坏模式；含初始裂纹铝合金板；冲击试验；数值仿真

［4］ Zhang M L，Chen Z，Zhao S. The determination of ele⁃ment failure strain with the element removal method［J］. Chinese Journal of Ship Research，2013，8（2）：

46-52，64（in Chinese）.［5］ Fan Z X，Li Y Z，Wang Y X，et al. Study on fracture

toughness and residual strength of aluminum alloy thinsheet with crack［J］. Advances in Aeronautical Sci⁃ence and Engineering，2015，6（1）：52-52，63（inChinese）.

［6］ Yin X G，Guo S X. Dynamic reliability analysis ofstructures with initial cracks［J］. Engineering Mechan⁃ics，1995，12（2）：72-79（in Chinese）.

［7］ Paik J K，Kumar Y V S，Lee J M. Ultimate strength ofcracked plate elements under axial compression or ten⁃sion［J］. Thin-Walled Structures，2005，43（2）：

237-272.［8］ Paik J K. Residual ultimate strength of steel plates with

longitudinal cracks under axial compression-experi⁃ments［J］. Ocean Engineering，2008，35（17-18）：

1775-1783.

［9］ Paik J K. Residual ultimate strength of steel plates withlongitudinal cracks under axial compression-nonlinearfinite element method investigations［J］. Ocean Engi⁃neering，2009，36（3/4）：266-276.

［10］ Zhang J，Yan Y，Xu X X，et al. Influence of cracksand pitting corrosion on residual ultimate strength ofstiffened plates［J］. Chinese Journal of Ship Re⁃search，2018，13（1）：38-45（in Chinese）.

［11］ Seifi R，Khoda-yari N. Experimental and numericalstudies on buckling of cracked thin-plates under fulland partial compression edge loading ［J］.Thin-Walled Structures，2011，49（12）：1504-1516.

［12］ Margaritis Y，Toulios M. The ultimate and collapseresponse of cracked stiffened plates subjected to uni⁃axial compression ［J］. Thin-Walled Structures，2012，50（1）：157-173.

［13］ Xu M C，Garbatov Y，Soares C G. Residual ultimatestrength assessment of stiffened panels with lockedcracks［J］. Thin-Walled Structures，2014， 85：398-410.

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