research article analysis on factors affecting the self

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 138162 6 pageshttpdxdoiorg1011552013138162

Research ArticleAnalysis on Factors Affecting the Self-Repair Capability ofSMA Wire Concrete Beam

Li Sun Dezhi Liang Qianqian Gao and Jianhong Zhou

School of Civil Engineering Shenyang Jianzhu University Shenyang 110168 China

Correspondence should be addressed to Li Sun sunli2009163com

Received 24 September 2013 Accepted 21 November 2013

Academic Editor Xiao-Wei Ye

Copyright copy 2013 Li Sun et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Crack expansion of concrete is the initial damage stage of structures whichmay cause greater damage to structures subject to long-term loads or under extreme conditions In recent years the application of intelligent materials to crack self-repair has become ahotspot among researchers In order to study the influence of factors on the self-repair capability of shape memory alloy (SMA)wire concrete beam both theoretical and experimental methods were employed for analysis For the convenience of experimentcompositematerials (epoxy cementmortar and silicone polymer clay) instead of concrete were usedThe SMAwires were externallyinstalled on and internally embedded in epoxy resin cement mortar beams and silicone polymer clay beams Comparison of crackrepair situation between two installationmethods turns out that bothmethods possess their own advantages and disadvantages andshould be employed according to the actual situation The influence of unbonded length on the self-repair capability of embeddedtype SMA wire beams and the necessary minimum unbonded length to achieve self-repair function were studied The results stateclearly that the longer the unbonded length is the better the crack repair situation is

1 Introduction

SMA [1] is a kind of material with shape memory functionrealized by phase transition which is induced by stress andtemperature Plastic deformation will be generated whenSMA which is shaped under a high temperature is placedin a low or room temperature environment When theambient temperature rises to the critical temperature thatis phase transition temperature the deformation disappearsand SMA can be recovered to the initial design state In thisprocess the displacement or stress in SMA is the function oftemperature The phenomenon that deformation disappearswith the increase of temperature and the shape recovers iscalled the shape memory effect

For concrete structures which are prone to crack theshape memory effect of SMA can be employed to reinforce orrepair the cracked structures One way is to fix SMAmaterialon the structure and the other is to embed SMA material asreinforcement in the structure reserving a certain unbondedlength

Many researchers have done much work in this field Byfixing SMA wires on concrete beams Luo [2] investigated

the driving performance and the repair performance of SMAwires By embedding prestressed SMA wires in concretebeams He et al [3] studied the response of the beams understatic loading and constant impact loading and the effectsof the SMA wires on the repair of cracks Embedding SMAwires in composite components Tao [4] tested the repairfunction of restoring force induced by heating in SMA wiresMaji and Negret [5] embedded SMA sterepsinema in mortartrabeculae and loaded used three-point bending test untilvisible cracks appeared Then the SMA sterepsinema washeated by electricity After a period of time the visible crackfully recovers which turns out that the SMA sterepsinemahasa great driving function

Shi Yan et al [6] applied the shape memory effect ofthe SMA to make intelligent reinforced concrete continuousbeams The principle of self-repairing property and thefactors affecting self-repairing property were experimentallystudied It turns out that deformation and crack width can besignificantly decreased by the heated SMA bars The drivingeffect can be enhanced by increasing the total cross-sectionarea of the SMA bars or the reinforcement ration of theSMA bars Li et al [7ndash9] investigated SMA for intelligently

2 Mathematical Problems in Engineering

Table 1 Properties of tested beams

Beam no Specimendimension (mm)

Unbonded lengthof beams 119897

1(mm)

Diameter of Ni-Tialloy wire (mm)

Thickness ofprotective layer

(mm)Type of material

Al-1 20 times 30 times 200 0 05 5 Mixtures of epoxy resins andcement mortars



Al-4 20 times 30 times 200 mdash mdash mdash Mixtures of epoxy resins andcement mortars

Bl-1 20 times 30 times 200 0 05 5 Mixtures of silicones and claysBl-2 20 times 30 times 200 10 05 5 Mixtures of silicones and claysBl-3 20 times 30 times 200 40 05 5 Mixtures of silicones and claysBl-4 20 times 30 times 200 mdash mdash mdash Mixtures of silicones and claysNote ldquomdashrdquo indicates that there is no SMA wire in the beam

self-repairing of concrete structures The results present thatthe heating method is an important influencing factor in therestoring force of SMA

However no further research has been done by scholarson comparison of the advantages and disadvantages betweenthe two installation methods mentioned above and the influ-ence of unbonded length on the repair capacity of internallyembedded installation method This is just the main subjectwhich the paper focuses on

2 Experiment Scheme

For the convenience of experiment the tested beams aremade from composite materials instead of concrete Sevengroups of beams numbered as Al-1 Al-2 Al-3 Al-4 Bl-1Bl-2 and Bl-3 are made Beams from Al-1 to Al-4 are madefrom epoxy resins and cement mortars and the rest are madefrom silicones and clays The dimension of beams and SMAalloy wires and the unbonded length of SMA alloy wires aretabulated in Table 1

The experiment was completed in the mechanical labora-tory of Shenyang Jianzhu University and a universal testingmachine was used as the loading device Firstly three-pointbending test was employed to load each beam until visiblecracks appeared The cracked beams were repaired in twoways One way was to internally embed SMA wires in thestructure with reserving a certain unbonded length as shownin Figure 1(a) The power was connected to the beam at bothends of the alloy wire for heating the alloy wire so that thecracks were repaired by force generated by heated alloy wireThis method was applied to beams numbered as Al-1 Al-2Al-3 BL-1 Bl-2 and Bl-3 in which the unbonded lengths ofSMA wires were different The other way was to externallyinstall Ni-Ti alloy wires on the surface of beam numberedas Al-4 as shown in Figure 1(b) The power was connectedto the beam at both ends of the alloy wire for heating thealloy wire so that the cracks were repaired by force generatedby heated alloy wire According to crack recovery situationcurrent strength and power-on time the advantages and

disadvantages of the two installation methods mentionedabove were compared and the influence of unbonded lengthof the internally embedded installation repair method wasanalysed

3 Factors Affecting the Self-RepairCapability of Polymer Beams

31 Influence of Installation Method As can be seen fromFigure 1 different bending moment results from differentacting location of the restoring force generated by the alloywire using the two methods Assume that the central axis isin the middle of the beam and mechanics is analyzed basedon the theory of the concrete structure

(1) Force analysis of internal embedded repair method

119872119898= 119865(ℎ

2minus 119888) (1)

119865 = 120590119878 (2)

where119872119898indicates the restoring bending moment 119865 is the

restoring force ℎ is the height of beam 119888 is the thickness ofthe protective layer 120590 is restoring stress in the SMAwire and119878 is the cross sectional area of the SMA wire

(2) Force analysis of external installation repair method

119872119891= 119865(ℎ

2+ 119909) (3)

where 119909 indicates the distance from the SMA wire to thebottom of the beam

Comparing (1) with (3)119872119898lt 119872119891if the restoring force

119865 is the same In view of this external installed SMA wire ismore favorable in beam crack closure

Experiment configurations of beam crack closure usingdifferent installationmethods of SMAwires are shown in Fig-ures 2 and 3 Bending moments in two cases are different dueto different acting points of the restoring forces 119865 generated

Mathematical Problems in Engineering 3

C

SMA alloy wire

Central axisF F h2

h

(a) Internal embedment repair method

Central axis

F F

SMA alloy wire Clamp for SMA wire installation

x

h2 h

(b) External installation repair method

Figure 1 Methods of repairing beam cracks

Table 2 The basic situations in beam crack closure processes

Beam no Crack width beforepower-on 119908 (mm)

Crack recovery valueΔ119908 (mm)

Crack recovery degree120578 ()

Current intensity119868 (A) Power-on time 119905 (s)

Al-1 3075 111 361 3 30Al-4 2212 2186 988 28 20

F

x1

Figure 2 External installed SMA wires

F

x2

Figure 3 Internal embedded SMA wires

by the alloy wires If restoring forces 119865 are the same we have1199091gt 1199092according to (1) and (3) Therefore the bending

moment in Figure 2 is larger than that in Figure 3 The crackcan be more easily recovered using external installed SMAwire (as shown in Figure 2) and less energy is consumedWithout regard to any other factor except for installationmethod comparison between two installation methods istabulated in Table 2

As can be seen from Table 2 external installed SMA wirenot only needs smaller current intensity and less power-ontime but also recovers much more than internal embeddedSMA wire does

C

SMA alloy wire

h0 h

w

w

0

Figure 4 Schematic diagramof internally embedded SMAwires forcrack closure

However the installation process of external installationmethod is more complicated There are not only preem-bedded parts in the structure but also installation devicesexternally erected outside the structure The cost of externalinstallation method is much higher than that of internallyembeddedmethod But SMAwireswhich are used in externalinstallation method are reusable and have lower cost andhigher activation force Therefore both methods have theirown advantages and disadvantagesWe should select the nec-essary installation method according to a specific situation

32 Influence of Unbounded Length in the Embedded SMABeam In order to study more accurately the influence ofunbounded length of internal embedded SMA wires onthe self-repairing capacity of beam the analysis method ofcontrolling crack width in reinforced concrete structures isreferred to According to the bond slip theory the actualoperative length of SMA alloy wires can be expressed as

119897 = 1199080+ 2 times 20119863 + 119897

1 (4)

where 119863 is the diameter of SMA wire 1198971is the unbounded

length of SMAwire and1199080is the length of SMAwire between

the surfaces of the crack as shown in Figure 4 20119863 is the sliplength of the SMA wire at both ends of the crack which isestimated according to the reinforcement anchorage lengthranges from concrete code [10] where the effective height ofthe cross-section of the beam is given by

ℎ0= ℎ minus 119888 (5)


Table 3 Theoretical recovered length of internally embedded SMA alloy wires

Beam no 119908 (mm) 1199080(mm) 119863 (mm) 119897

1(mm) 119897 (mm) 120576 () 120575 (mm)

Al-1 3075 2561 005 0 22561 5 1128Al-2 2842 2367 005 10 32367 5 1618Al-3 2975 2478 005 40 62478 5 3124Bl-1 2865 2387 005 0 22387 5 1119Bl-2 4514 3760 005 10 33760 5 1688Bl-3 2681 2233 005 40 62233 5 3112

Table 4 Experimental and theoretical values of crack recovery

Beam no Crack width119908 (mm)

Experimental value Theoretical valueValues of crack recoveryΔ119908 (mm)

Degree of crack recovery120578 ()

Values of crack recoveryΔ1199081015840 (mm)


Al-1 3075 111 361 1354 550Al-2 2842 1828 643 1943 316Al-3 2975 2879 968 2975 100Bl-1 2865 1123 392 1344 531Bl-2 4514 325 720 2026 551Bl-3 2681 2617 976 2681 100

According to the proportional relationship in Figure 4

1199080=ℎ0

ℎ119908 (6)

By introducing (5) into (6) the following equation can beobtained

1199080=ℎ minus 119888

ℎ119908 (7)

where ℎ is the height of beam 119888 is the thickness of theprotective layer and 119908 is the crack width By introducing(7) into (4) the actual operative length of alloy wire can beobtained when the unbonded length of SMA wire is 119897

1

119897 =ℎ minus 119888

ℎ119908 + 2 times 20119863 + 119897

1 (8)

If the strain of SMAwire is 120576 and the SMAwire completelyrecovers its initial shape the recovery strain is also 120576 Whenthe unbounded length of SMA wire is 119897

1and the cracks are

completely repaired the recovered length of alloy wire can beobtained as

120575 = 119897120576 = (ℎ minus 119888

ℎ119908 + 2 times 20119863 + 119897

1) 120576 (9)

By substituting the parameters of internally embeddedSMA wires beam into (9) we can obtain the value of thetheoretically recovered length of SMA wires as shown inTable 3

Based on the unbonded length of SMA wires tabulatedin Table 3 the experiment study on the self-repair ability ofbeam cracks was conducted Figure 5 shows the influence ofunbonded length of SMA wires on recovery degree of cracks

Beams numbered as A1-1 Al-2 and Al-3 are made ofthe same materials As can be seen from Figure 5 andTable 3 the longer the unbounded length is the better thecracks recover Cracks on beam Al-3 are almost nonvisibleTherefore the unbonded length of SMA wires in beam Al-3meets the required length Similar situations for beams Bl-1Bl-2 and Bl-3 can be observed as shown in Figure 5

However further study is necessary to verify non-theoretical unbonded length of SMA wires in the beamthat meets the required length for repairing cracks Theexperimental and theoretical variations of crack width dur-ing recovery process are tabulated in Table 4 where theunbonded length in beams A1-1 Al-2 and Al-3 are respec-tively 0mm 10mm and 40mm and the same in beams B1-1Bl-2 Bl-3

As can be seen from Table 4 the experimental andtheoretical values of crack recovery in beams with internallyembedded SMA wires are in good agreement Thereforein order to achieve the self-repair ability the unbondedlength 119897

119898in embedded SMA polymer beams must satisfy

the relationship 119897119898ge 1198971 The minimum unbonded length

achieving self-repair is given by

119897119898ge 1198971=120575

120576minusℎ minus 119888

ℎ119908 minus 2 times 20119863 (10)

where 120575 represents the recovered length of SMA wire 120576 is therecovery strain of SMA wire ℎ is the height of beam 119888 is thethickness of the protective layer and119863 is the diameter of SMAwire

33 Other Factors Affecting Self-Repairing Capability Thematerial characteristics of the beams also affect the self-repairing capability The stiffness of beam Bl-3 made frompouring clays and silicones is small More energy is needed


The c

rack

reco

very

situ

atio

n of

bea

ms w

ith d

iffer

ent b

ond

leng

th o

f allo

y w

ires

The crack recovery situation of beams made of different materials

Beam Bl-1Beam Al-1

Beam Bl-2Beam Al-2

Beam Bl-3Beam Al-3

Figure 5 Crack recovery situation of beams with internally embedded SMA wires

Figure 6 Crack closure of beam Al-3

to recover the deformation The stiffness of beam Al-3made from pouring epoxy cements and mortars is large andthe energy required for repairing is relatively less Under thecondition of very nearly the same recovery degree in beamsBl-3 and A1-3 as shown in Figure 6 and Figure 7 and thesame power-on time the required current intensity for beamA1-3 is 40 percent less than that for beam B1-3

The current intensity and the crack closure situation aretabulated in Table 5 As can be seen the required currentintensity for the beammade from silicones and clays is larger

Figure 7 Crack closure of beam Bl-3

than that for the beam made from epoxy resins and cementmortars Therefore the material characteristics do not play adecisive role in crack closure of beams but they have certaininfluence on it Repairing efficiency of the beam made fromflexible materials is low and the current intensity must belarge enough to achieve full recovery

The shape memory effect and superelastic characteristicsof SMAmaterials of different types are different and the bondbetween bars and sterepsinema with aggregate in the beamis also different Therefore the properties of SMA alloy alsoaffect the self-repair ability In addition the beam crack slags


Table 5 Current intensity and crack closure degree

Beamno

Crackwidth 119908(mm)

Values ofcrack recoveryΔ119908 (mm)

Degree ofcrack recovery120578 ()

Currentintensity 119868 (A)

Al-1 3075 111 361 3Al-2 2842 1828 643 3Al-3 2975 2879 968 3Al-4 2212 2186 988 28Bl-1 2865 1123 392 5Bl-2 4514 325 720 5Bl-3 2681 2617 976 5Bl-4 4012 4005 998 35

will be stuck in the cracks andwil hinder crack closure whichalso has influence on crack closure

4 Conclusion

In this paper based on the small stiffness of the compositematerials and obvious crack closure phenomenon the crackclosure capacity of beams is theoretically and experimentallyanalyzed and it can be concluded as follows

(1) External installation method and internal embed-ment method are viable in crack closure and bothmethods have their own advantages and disadvan-tages We should select the appropriate methodaccording to specific conditions

(2) The minimum unbonded length of the embeddedSMA polymer beams that achieve repair is given by(10)

(3) Properties of beam material and SMA affect crackclosure These factors should be avoided or excludedin the repairing process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported by the National Natu-ral Science Foundation of China (51178277) Program forNew Century Excellent Talents in University (NCET-12-1013) Program for Liaoning Excellent Talents in University(LR2012018) and the Shenyang City Science Foundation ofChina (F11-163-9-00)

References

[1] Z-P Cao and S-L Wang ldquoStudy on application of intelligentmaterial systems and structures to civil engineeringrdquo Journal ofChongqing Jianzhu University vol 23 no l pp 1ndash4 2001

[2] Z-W Luo Basic Theoretical Studies on Intelligent ConcreteBeams Using Unbonded External SMAWires Tongji UniversityShanghai China 2006

[3] S-L He X-Y Huang and M-S Chen ldquoThe research of shapememory alloy reinforced concrete with self-diagnosis and self-adaptation smart structuresrdquo in Proceedings of the ChineseMaterials Symposium pp 622ndash625 Industry Press BeijingChina 1996

[4] B-Q Tao Smart Materials Structures Defense Industry Press1997

[5] A K Maji and I Negret ldquoSmart prestressing with shape-memory alloyrdquo Journal of Engineering Mechanics vol 124 no10 pp 1121ndash1128 1998

[6] S Yan J Sun and W Wang ldquoDeformation property of smartconcrete continuous beams reinforcedwith shapememory alloyrebarsrdquo Concrete pp 8ndash12 2010

[7] H Li C X Mao Z Q Liu and J P Ou ldquoShape memory alloy-based smart civil structures with self-sensing and repairingcapabilitiesrdquo in Sensing Issues in Civil Structural Health Moni-toring Chapter 4 pp 259ndash268 2005

[8] H Li Z-Q Liu and J-P Ou ldquoExperimental study of a simplereinforced concrete beam temporarily strengthened by SMAwires followed by permanent strengthening with CFRP platesrdquoEngineering Structures vol 30 no 3 pp 716ndash723 2008

[9] H Li Z-Q Liu and J-P Ou ldquoStudy on reinforced concretebeams strengthened using shape memory alloy wires in com-bination with carbon-fiber-reinforced polymer platesrdquo SmartMaterials and Structures vol 16 no 6 pp 2550ndash2559 2007

[10] D-J Ding Modern Concrete Structures China ArchitecturalPress 2000

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Table 1 Properties of tested beams

Beam no Specimendimension (mm)

Unbonded lengthof beams 119897

1(mm)

Diameter of Ni-Tialloy wire (mm)

Thickness ofprotective layer

(mm)Type of material




Al-4 20 times 30 times 200 mdash mdash mdash Mixtures of epoxy resins andcement mortars

Bl-1 20 times 30 times 200 0 05 5 Mixtures of silicones and claysBl-2 20 times 30 times 200 10 05 5 Mixtures of silicones and claysBl-3 20 times 30 times 200 40 05 5 Mixtures of silicones and claysBl-4 20 times 30 times 200 mdash mdash mdash Mixtures of silicones and claysNote ldquomdashrdquo indicates that there is no SMA wire in the beam

self-repairing of concrete structures The results present thatthe heating method is an important influencing factor in therestoring force of SMA

However no further research has been done by scholarson comparison of the advantages and disadvantages betweenthe two installation methods mentioned above and the influ-ence of unbonded length on the repair capacity of internallyembedded installation method This is just the main subjectwhich the paper focuses on

2 Experiment Scheme

For the convenience of experiment the tested beams aremade from composite materials instead of concrete Sevengroups of beams numbered as Al-1 Al-2 Al-3 Al-4 Bl-1Bl-2 and Bl-3 are made Beams from Al-1 to Al-4 are madefrom epoxy resins and cement mortars and the rest are madefrom silicones and clays The dimension of beams and SMAalloy wires and the unbonded length of SMA alloy wires aretabulated in Table 1

The experiment was completed in the mechanical labora-tory of Shenyang Jianzhu University and a universal testingmachine was used as the loading device Firstly three-pointbending test was employed to load each beam until visiblecracks appeared The cracked beams were repaired in twoways One way was to internally embed SMA wires in thestructure with reserving a certain unbonded length as shownin Figure 1(a) The power was connected to the beam at bothends of the alloy wire for heating the alloy wire so that thecracks were repaired by force generated by heated alloy wireThis method was applied to beams numbered as Al-1 Al-2Al-3 BL-1 Bl-2 and Bl-3 in which the unbonded lengths ofSMA wires were different The other way was to externallyinstall Ni-Ti alloy wires on the surface of beam numberedas Al-4 as shown in Figure 1(b) The power was connectedto the beam at both ends of the alloy wire for heating thealloy wire so that the cracks were repaired by force generatedby heated alloy wire According to crack recovery situationcurrent strength and power-on time the advantages and

disadvantages of the two installation methods mentionedabove were compared and the influence of unbonded lengthof the internally embedded installation repair method wasanalysed

3 Factors Affecting the Self-RepairCapability of Polymer Beams

31 Influence of Installation Method As can be seen fromFigure 1 different bending moment results from differentacting location of the restoring force generated by the alloywire using the two methods Assume that the central axis isin the middle of the beam and mechanics is analyzed basedon the theory of the concrete structure

(1) Force analysis of internal embedded repair method

119872119898= 119865(ℎ

2minus 119888) (1)

119865 = 120590119878 (2)

where119872119898indicates the restoring bending moment 119865 is the

restoring force ℎ is the height of beam 119888 is the thickness ofthe protective layer 120590 is restoring stress in the SMAwire and119878 is the cross sectional area of the SMA wire

(2) Force analysis of external installation repair method

119872119891= 119865(ℎ

2+ 119909) (3)

where 119909 indicates the distance from the SMA wire to thebottom of the beam

Comparing (1) with (3)119872119898lt 119872119891if the restoring force

119865 is the same In view of this external installed SMA wire ismore favorable in beam crack closure

Experiment configurations of beam crack closure usingdifferent installationmethods of SMAwires are shown in Fig-ures 2 and 3 Bending moments in two cases are different dueto different acting points of the restoring forces 119865 generated


C

SMA alloy wire

Central axisF F h2

h


Central axis

F F


x

h2 h








Al-1 3075 111 361 3 30Al-4 2212 2186 988 28 20

F

x1


F

x2





C

SMA alloy wire

h0 h

w

w

0




119897 = 1199080+ 2 times 20119863 + 119897

1 (4)




ℎ0= ℎ minus 119888 (5)



Beam no 119908 (mm) 1199080(mm) 119863 (mm) 119897

1(mm) 119897 (mm) 120576 () 120575 (mm)

Al-1 3075 2561 005 0 22561 5 1128Al-2 2842 2367 005 10 32367 5 1618Al-3 2975 2478 005 40 62478 5 3124Bl-1 2865 2387 005 0 22387 5 1119Bl-2 4514 3760 005 10 33760 5 1688Bl-3 2681 2233 005 40 62233 5 3112







Al-1 3075 111 361 1354 550Al-2 2842 1828 643 1943 316Al-3 2975 2879 968 2975 100Bl-1 2865 1123 392 1344 531Bl-2 4514 325 720 2026 551Bl-3 2681 2617 976 2681 100


1199080=ℎ0

ℎ119908 (6)


1199080=ℎ minus 119888

ℎ119908 (7)


1

119897 =ℎ minus 119888

ℎ119908 + 2 times 20119863 + 119897

1 (8)


1and the cracks are


120575 = 119897120576 = (ℎ minus 119888

ℎ119908 + 2 times 20119863 + 119897

1) 120576 (9)









119897119898ge 1198971=120575


ℎ119908 minus 2 times 20119863 (10)




The c

rack

reco

very

situ

atio

n of

bea

ms w

ith d

iffer

ent b

ond

leng

th o

f allo

y w

ires


Beam Bl-1Beam Al-1

Beam Bl-2Beam Al-2

Beam Bl-3Beam Al-3










Beamno







4 Conclusion







Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










C

SMA alloy wire

Central axisF F h2

h


Central axis

F F


x

h2 h








Al-1 3075 111 361 3 30Al-4 2212 2186 988 28 20

F

x1


F

x2





C

SMA alloy wire

h0 h

w

w

0




119897 = 1199080+ 2 times 20119863 + 119897

1 (4)




ℎ0= ℎ minus 119888 (5)



Beam no 119908 (mm) 1199080(mm) 119863 (mm) 119897

1(mm) 119897 (mm) 120576 () 120575 (mm)

Al-1 3075 2561 005 0 22561 5 1128Al-2 2842 2367 005 10 32367 5 1618Al-3 2975 2478 005 40 62478 5 3124Bl-1 2865 2387 005 0 22387 5 1119Bl-2 4514 3760 005 10 33760 5 1688Bl-3 2681 2233 005 40 62233 5 3112







Al-1 3075 111 361 1354 550Al-2 2842 1828 643 1943 316Al-3 2975 2879 968 2975 100Bl-1 2865 1123 392 1344 531Bl-2 4514 325 720 2026 551Bl-3 2681 2617 976 2681 100


1199080=ℎ0

ℎ119908 (6)


1199080=ℎ minus 119888

ℎ119908 (7)


1

119897 =ℎ minus 119888

ℎ119908 + 2 times 20119863 + 119897

1 (8)


1and the cracks are


120575 = 119897120576 = (ℎ minus 119888

ℎ119908 + 2 times 20119863 + 119897

1) 120576 (9)









119897119898ge 1198971=120575


ℎ119908 minus 2 times 20119863 (10)




The c

rack

reco

very

situ

atio

n of

bea

ms w

ith d

iffer

ent b

ond

leng

th o

f allo

y w

ires


Beam Bl-1Beam Al-1

Beam Bl-2Beam Al-2

Beam Bl-3Beam Al-3










Beamno







4 Conclusion







Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Beam no 119908 (mm) 1199080(mm) 119863 (mm) 119897

1(mm) 119897 (mm) 120576 () 120575 (mm)

Al-1 3075 2561 005 0 22561 5 1128Al-2 2842 2367 005 10 32367 5 1618Al-3 2975 2478 005 40 62478 5 3124Bl-1 2865 2387 005 0 22387 5 1119Bl-2 4514 3760 005 10 33760 5 1688Bl-3 2681 2233 005 40 62233 5 3112







Al-1 3075 111 361 1354 550Al-2 2842 1828 643 1943 316Al-3 2975 2879 968 2975 100Bl-1 2865 1123 392 1344 531Bl-2 4514 325 720 2026 551Bl-3 2681 2617 976 2681 100


1199080=ℎ0

ℎ119908 (6)


1199080=ℎ minus 119888

ℎ119908 (7)


1

119897 =ℎ minus 119888

ℎ119908 + 2 times 20119863 + 119897

1 (8)


1and the cracks are


120575 = 119897120576 = (ℎ minus 119888

ℎ119908 + 2 times 20119863 + 119897

1) 120576 (9)









119897119898ge 1198971=120575


ℎ119908 minus 2 times 20119863 (10)




The c

rack

reco

very

situ

atio

n of

bea

ms w

ith d

iffer

ent b

ond

leng

th o

f allo

y w

ires


Beam Bl-1Beam Al-1

Beam Bl-2Beam Al-2

Beam Bl-3Beam Al-3










Beamno







4 Conclusion







Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










The c

rack

reco

very

situ

atio

n of

bea

ms w

ith d

iffer

ent b

ond

leng

th o

f allo

y w

ires


Beam Bl-1Beam Al-1

Beam Bl-2Beam Al-2

Beam Bl-3Beam Al-3










Beamno







4 Conclusion







Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Beamno







4 Conclusion







Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article analysis on factors affecting the self

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