Research Article Study of the Residual Strength of an RC

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 101484 14 pageshttpdxdoiorg1011552013101484

Research ArticleStudy of the Residual Strength of an RC Shear Wall with FractalCrack Taking into Account Interlocking Interface Phenomena

O Panagouli and K Iordanidou

Laboratory of Structural Analysis and Design Department of Civil Engineering University of Thessaly Pedion Areos38334 Volos Greece

Correspondence should be addressed to O Panagouli olpanaguthgr

Received 10 July 2013 Revised 25 September 2013 Accepted 2 October 2013

Academic Editor Anaxagoras Elenas

Copyright copy 2013 O Panagouli and K Iordanidou This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In the present paper the postcracking strength of an RC shear wall element which follows the construction practices applied inGreece during the 70s is examined by taking into account the complex geometry of the crack of the wall and the mixed friction-plastification mechanisms that develop in the vicinity of the crack Due to the significance of the crack geometry a multiresolutionanalysis based on fractal geometry is performed taking into account the size of the aggregates of concrete The materials (steeland concrete) are assumed to have elastic-plastic behaviour For concrete both cracking and crushing are taken into account in anaccurate manner On the interfaces of the crack unilateral contact and friction conditions are assumed to hold For every structurecorresponding to each resolution of the interface a classical Euclidean problem is solved The obtained results lead to interestingconclusions concerning the influence of the simulation of the geometry of the fractal crack on the mechanical interlock betweenthe two faces of the crack a factor which seems to be very important to the postcracking strength of the lightly reinforced shearwall studied here

1 Introduction

Many RC structures are facing a number of challenges thatis earthquakes hurricanes and so forth which may threatentheir safety and serviceability Therefore modern structuresbuilt in seismic prone areas are designed to have significantbending and shear strength and ductility However existingstructures designed according to earlier versions of theseismic codes and constructed using low strength materialsusually have inadequate shear strength For that shear cracksappear in the shear wall elements of these structures reducingtheir overall capacity

Generally cracks are of large interest in RC structuressince their properties reflect not only the condition ofconcrete as material but also the condition of the entiresystem at structural level Crack width is commonly usedas a convenient indicator of damage to RC elements but itshould be noted that the distribution and the geometry of thecracks are also important in measuring the extent of damage

presented in the structure [1 2] and in calculating the residualstrength of it

It is well known that the geometry of the interfaces of acrack is of fundamental importance to the study of frictionwear and also strength evaluation The recent research offractured surfaces of various materials provides a deeperinsight into the geometry of cracks The correspondingresearch on metals [3] and on concrete and rocks [4ndash8]showed that the fractured surfaces hold fractal propertiesin a well-defined scale range In the case of concrete theeffects of aggregates sizes and the quality of concrete on thefractality of fractured surfaces were also investigated in [9ndash12] respectively Therefore an accurate description of thegeometry of crack interfaces is of great importance for thesimulation of contact It is important to mention here thatthe actual contact between two real interfaces is realizedonly over a small fraction in a discrete number of areasConsequently the real contact area is only a fraction ofthe apparent area [13 14] and the parameters of the actual

2 Mathematical Problems in Engineering

contact regions are strongly influenced by the roughness ofthe contacting surfaces

The multiscale nature of the surface roughness suggeststhe use of fractal geometryThe fractal approach adopted herefor the simulation of the geometry of the cracks formed ina shear wall uses computer generated self-affine curves forthe modelling of the interface roughness which is stronglydependent on the values of the structural parameters ofthese curves The computer generated fractal interfaceswhich are based on a given discrete set of interface dataare characterized by a precise value of the resolution 120575 ofthe interface This fact permits the study of the interfaceroughness on iteratively generated rough profiles makingthis approach suitable for engineering problems since itpermits the satisfactory study of the whole problem withreliable numerical calculations

The aim of this paper is to study how the resolutionof a fractal crack F affects the strength of a reinforcedconcrete shear wall element On the interface between thetwo cracked surfaces unilateral contact and friction con-ditions are assumed to hold The applied approach takesinto account the nonlinear behaviour of the materialsincluding the limited strength of concrete under tensionThe shear wall is submitted to shear loading As a resultof the applied approach the contribution of the frictionbetween the cracked surfaces is taken into account as wellas the additional strength coming from the mechanicalinterlock between the two faces of the crack For everystructure resulting for each resolution of the interface aclassical Euclidean problem is solved by using a variationalformulation [15] It must be mentioned here that the finestresolution of the interface is related to the size of theaggregates

2 Fractal Representation of Rough Surfaces

The fractal nature of material damage has been a matter ofa very intense research during the last three decades Thefractal nature of fractured surfaces in metals was shownmore than 30 years ago by Mandelbrot et al [3] Morespecifically it was shown that the fractured surfaces inmetals develop a fractal structure over more than threeorders of magnitude In quasibrittle materials observationshave shown that fractured surfaces display self-affine scaleproperties in a certain range of scales which is in mostcases very large and which greatly depends on the materialmicrostructure This is true for a large variety of quasibrittlematerials such as rock concrete and ceramics [4ndash8 16ndash18]

Fractal sets are characterized by noninteger dimensions[19] The dimension of a fractal set in plane can vary from0 to 2 Accordingly by increasing the resolution of a fractalset its length tends to 0 if its dimension is smaller than 1(totally disconnected set) or tends to infinity if it is largerthan 1 In these cases the length is a nominal useless quantitysince it changes as the resolution increases Converselythe fractal dimension of a fractal set is a parameter of

great importance because of its scale-independent charac-ter

Many methods which are based on experimental ornumerical calculations such as the Richardson method[19] have been developed for the estimation of the fractaldimension of a curve According to this method dividerswhich are set to a prescribed opening120575 are usedMovingwiththese dividers along the curve so that each new step startswhere the previous step leaves off one obtains the numberof steps 119873(120575) The curve is said to be of fractal nature if byrepeating this procedure for different values of 120575 the relation

119873(120575) sim 120575minus119863 (1)

is obtained in some interval 120575lowast lt 120575 lt Δlowast The power 119863denotes the fractal dimension of the profile which is in therange 1 lt 119863 lt 2 The relation between the fractal dimension119863 of this profile and the dimension of the correspondingsurface is119863

119904

= 119863 + 1 [16]In relation (1) there is an upper and a lower bound

in the scaling range and consequently a transition fromthe fractal regime at the microscopic level to the Euclideanregime at largest scales The upper bound is representedby the macroscopic size of the set while the lower oneis related to the size of the smallest measurable particlesthat is the aggregates in the case of concrete Mandelbrot[20] first pointed out the transition from a fractal regimecharacterized by noninteger dimensions to the homogeneousone characterized by classical topological dimensions a factwhich points out the main difference between mathematicaland physical fractals

The idea of self-affinity is very popular in studying surfaceroughness because experimental studies have shown thatunder repeated magnifications the profiles of real surfacesare usually statistically self-affine to themselves [3 21] Theself-affine fractals were used in a number of papers as atool for the description of rough surfaces [22ndash28] Typicallysuch a profile can be measured by taking height data 119910

119894

with respect to an arbitrary datum at 119873 equidistant discretepoints 119909

119894

In the sequence fractal interpolation functionsF(119909119894

) = 119910119894

119894 = 0 1 119873 are used for the passage fromthis discrete set of data (119909

119894

119910119894

) 119894 = 0 1 2 119873 to acontinuous model According to the theory of Barnsley [29]the sequence of functionsF

119899+1

(119909) = (119879F119899

)(119909) = 119888119894

119897119894

minus1

(119909) +

119889119894

F119899

(119897119894

minus1

(119909)) + 119892119894

converges to a fractal curveF as 119899 rarr infinThe transformation 119897

119894

transforms [1199090

119909119873

] to [119909119894minus1

119909119894

] and itis defined by the relation 119897

119894

(119909) = 119886119894

119909 + 119887119894

The calculation ofthe parameters 119886

119894

119887119894

119888119894

and119892119894

is based on the given set of dataand the free parameters 119889

119894

Fractal interpolation functions give profiles which look

quite attractive from the viewpoint of a graphic roughnesssimulation In higher approximations these profiles appearrougher as it is shown in the next section where the first tofifth approximations of a fractal interpolation function arepresented Moreover the roughness of the profile is stronglyaffected by the free parameters 119889

119894

of the interpolation func-tions As these parameters take larger values the resultingprofiles appear rougher with sharper peaks

Mathematical Problems in Engineering 3

qN

P 120575

Fixed boundary

1500

2950

y

x 500

4000

4500

4empty20empty82004empty20

Figure 1 The considered shear wall

Another model of self-affine profiles which can be usedfor roughness description is the multilevel hierarchical pro-fileThis profile has a hierarchical structure and is constructedby using a certain iterative scheme presented in [24] As inthe case of the fractal interpolation functions the surfacesproduced by this scheme are characterized by a precise valueof the resolution 120575 of the fractal curve More specifically inboth cases the iterative construction of the profiles permits usto analyse ldquoprefractalsrdquo of arbitrary generation and thereforeof arbitrary resolution 120575

119899

It must be mentioned here that an important advantage

of the fractal interpolation functions presented in [29] andof the multilevel hierarchical approach presented in [24] isthat their fractal dimension can be obtained analytically andit depends strongly on their construction parameters Thusin the case of fractal interpolation functions which are usedin this paper for the simulation of the geometry of the crackthe fractal dimension119863 is given by the relation

119873

sum

119894=1

10038161003816100381610038161198891198941003816100381610038161003816 119886119863minus1

119894

= 1 (2)

3 Description of the Considered Problem

In Figure 1 an RC shear wall element which follows theconstruction practices applied in Greece during the 70s ispresentedMore specifically the wall is reinforced by a doublesteel mesh consisting of horizontal and vertical rebars havinga diameter of 8mm and a spacing of 200mm The quality ofthe steel mesh is assumed to be S220 (typical for buildingsof that age) At the two ends of the wall the amount ofreinforcement is higher Four 20mm rebars of higher quality(S400) are used without specific provisions to increase theconfinement The thickness of the wall is 200mm and thequality of concrete is assumed to be C16 typical for this kindof constructions The wall is fixed on the lower horizontalboundary

Table 1 Characteristics of the considered structures

Iteration (119899) Resolution 120575119899

(m) Interface length 119871119899

(m)1st 1404 48882nd 0468 49463rd 0156 50804th 0052 53735th 0017 5935

The considered shear wall is divided into two parts by acrack which has been formed due to shear failure of concreteIt is important to be mentioned here that in low strengthconcretes as in the case examined here the fractured surfacesare rougher compared to the fractured surfaces developedin high strength concretes [12] because in the first casethe cracks are developed in the contact zone between theaggregates and the cement paste whereas in the second casethe failure of the aggregates ensures a less rough interfaceFor the description of the roughness of the crack the notionof fractals is used More specifically the crack is describedby a fractal interpolation function which interpolates the setof data (minus10 295) (04 20) (18 10) (32 05) The freeparameters of the function are taken to have the values 119889

1

=

1198892

= 1198893

= 050 in order for the interface to be rough (thefractal dimension of the interface results to be equal to 1369)

The computer generated interfaces F119899

119899 = 1 2

are ldquoprefractalsrdquo images of the fractal set characterized bya precise value of the resolution 120575

119899

which is related tothe 119899th iteration of the fractal interpolation function andrepresents the characteristic linear size of the interface Asit is shown in Figure 2 where five iterations of a fractalinterface are given the linear size of the interface changesrapidly when higher iterations are taken into account InTable 1 the characteristics of each resolution are presentedMore specifically in the second column the resolution of theinterface 120575

119899

is given whereas in the third column the totalcrack length 119871

119899

is presentedThe objective here is to estimate the capacity of the shear

wall under an action similar to the one that has created thecrack For this reason a horizontal displacement of 20mm isapplied on the upper side of the wall (see Figure 1) Moreovera vertical distributed loading 119902

119873

is applied on the upperhorizontal boundary creating a compressive axial loadingThe resultant of this loading is denoted by 119873 For 119873 sixdifferent values will be considered from 0 to 2500 kN witha step of 500 kN

For the modelling of the above problem it is assumedthat the opposite sides of the fracture are perfectly matchingsurfaces in a distance of 01mm and the finite elementmethod is used In order to avoid a much more complicatedthree-dimensional analysis two-dimensional finite elementswere employed however special consideration was givento the incorporation of the nonlinearities that govern theresponse of the wall More specifically the mass of concretewas modelled through quadrilateral and triangular plainstress elements The finite element discretization density issimilar for all the considered problems [30] in order for thediscretization density not to affect the comparison between


1st resolution 2nd resolution

3rd resolution 4th resolution

5th resolution

Figure 2 The first five resolutions of the fractal crack

the results of the various analyses that were performed Themodulus of elasticity for the elements representing the massof concrete was taken to be equal to 119864 = 21GPa and thePoissonrsquos coefficient to be equal to ] = 016 The material wasassumed to follow the nonlinear law depicted in Figure 3(a)Under compression the material behaves elastoplasticallyuntil a total strain of 0004 After this strain value crushingdevelops in concrete leading its strength to zero A morecomplicated behaviour is considered under tension Morespecifically after the exhaustion of the tension strength of

concrete a softening branch follows having a slope 119896119904

=

10GPa Progressively the tension strength of concrete is alsozeroed The above unidirectional nonlinear law is comple-mented by an appropriate yield criterion (Tresca) which takesinto account the two-dimensional stress fields that developin the considered problem For the simulation of crackinga smeared crack algorithm is used in which the cracks areevenly distributed over the area of each finite element [31]

The steel rebars were modelled through two-dimensionalbeam elements which were connected to the same grid of


0004

136

16

Compression

Tension

120590 (MPa)

ks120576

(a)

220

02

264

minus220minus264

120590 (MPa)

120576ay

(b)

400

02

480

minus400minus480

120590 (MPa)

120576ay

(c)

Figure 3 The adopted materials laws (a) C16 concrete (b) S220 steel and (c) S400 steel

nodes as the plain stress elements simulating the concreteAt each position the properties that were given to the steelrebars take into account the reinforcement that exists in thewhole depth of the wall For example the horizontal andvertical elements that simulate the steel mesh are assigned anarea of 10048mm2 that corresponds to the cross-sectionalarea of two 8mm steel rebars For simplicity the edge rein-forcements (412060120) were simulated by a single row of beamelements that have an area of 1256mm2 (ie 4 times 314mm2)For the steel rebars a modulus of elasticity 119864 = 210GPa wasassumed Moreover the nonlinear laws of Figures 3(b) and3(c) which exhibit a hardening branch after the yield stressof thematerial is attained were considered for S220 and S400steel qualities respectively

Figure 4 depicts the finite element discretizations forthe structures that correspond to the third fourth andfifth iterations of the fractal crack The grey lines in thefinite element meshes correspond to the positions of thesteel rebars Special attention was given in the modellingso that the steel rebars retain their initial horizontal andvertical positions that is no eccentricity exists betweenthe corresponding rows of beam finite elements due to theformation of the crack

In this paper only the finite element models correspond-ing to the 3rd 4th and 5th resolutions of the fractal crack

were considered because 1st and 2nd resolutions do nothave a meaning from the engineering point of view On theother hand the 5th resolution gives a good lower bound of 120575because 120575

5

is related to the size of the aggregates of concreteAt the interfaces unilateral contact and friction condi-

tions were assumed to hold The Coulombrsquos friction modelwas followed with a coefficient equal to 06 At each scale aclassical Euclidean problem is solved by using a variationalformulation [15]

For every value of the vertical loading 119873 a solution istaken in terms of shear forces and horizontal displacementsat the interface for different values of the resolution of thecracked wall and for the case of the uncracked wall The aimof this work is to study the behaviour of the shear wall thatis the behaviour of concrete and the forces in the rods as thevertical loading and the resolution of the interface change

Two cases are considered

(i) In the first case the wall is uncracked

(ii) In the second case where a fractal crack F has beendeveloped in the wall different resolutions are takeninto account in order to examine how the resolutionof a fractal interfaceF affects the strength of the RCshear wall element


Figure 4 FE discretizations for third fourt and fifth resolutions of the fractal interface

The solution of the above problems is obtained throughthe application of the Newton-Raphson iterative methodDue to the highly nonlinear nature of the problem a veryfine load incrementation was used The maximum value ofthe horizontal displacement (20mm) was applied in 2000loading steps while the total vertical loading was applied inthe 1st load step and was assumed as being constant in thesubsequent steps

4 Numerical Results

Figure 5 presents the applied horizontal load versus thecorresponding displacement (119875-120575 curves) for the differentvalues of vertical loading 119873 Starting from the case of theuncracked wall it must be mentioned that the value ofthe vertical loading plays a significant role As the value ofthe vertical loading increases the capacity of the wall toundertake horizontal loading increases as well However forhigher load values (for 119873 = 2000 and 2500 kN) strengthdegradations due to the exhaustion of the shear strength ofconcrete are noticed In the sequel the resistance of the wall

increases again as a result of the transfer of the loading fromconcrete to the horizontal steel rebars

In cases of the cracked walls the beneficial effect of thenormal compressive loading is oncemore verifiedThis resultholds for the 3rd the 4th and the 5th resolutions of thefractal crack but for small displacement values only For largerdisplacement values the three variants of the cracked wallbehave differently The 4th and the 5th resolutions appearto have a stable behaviour without strength degradationsHowever it is noticed that in the case of the 3rd resolutionand for heavy axial loading significant strength degradationtakes place

The above results can be more easily understood if wecompare in the same diagram the curves obtained for the fourdifferent structures studied here (uncracked 3rd resolution4th resolution and 5th resolution) for specific load levelsIn Figure 6 the P-120575 curves for three cases of axial loading(119873 = 0119873 = 1500 kN and119873 = 2500 kN) are presented Weobserve that for low values of the compressive axial loadingthere is actually no difference between the uncracked andthe cracked walls In all cases the horizontal loading is easilytransferred and no signs of strength degradation are noticed


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

Uncracked wall

0

200

400

600

800

1000

1200

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

Cracked wall-3rd fractal approximation

0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

Cracked wall-4th fractal approximation

0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


N = 0

N = 500

N = 1000

N = 1500

N = 2000

N = 2500

Figure 5 Load-displacement (P-120575) curves for the cases of theuncracked and cracked walls

0

100

200

300

400

500

600

700

800

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

N = 0

0

200

400

600

800

1000

1200

1400

0 0005 001 0015 002

Forc

e (kN

)Fo

rce (

kN)

Displacement (m)

N = 1500

0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002Displacement (m)

N = 2500kN

UncrackedCracked-3rd iter

Cracked-4th iterCracked-5th iter

Figure 6 Comparison of the behaviour of the four variants of theexamined wall for specific values of the compressive axial loading

because thewall worksmainly in bending and the shear forcesare well below the shear strength of the wall For moderateaxial loading values (ie for119873 = 1500 kN) it is noticed thatthe uncracked wall appears to have greater strength than thecracked variants examined here It is also noticed that the5th resolution of the fractal crack leads to greater ultimatestrength a result which can be attributed to the fact that the5th resolution seems to lead to a greater degree of interlockingbetween the two interfaces of the crack

However the most interesting case is the case whereheavy axial loading (119873 = 2500 kN) is applied to the wallFirst it can be noticed that the behaviour of the 4th and


4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN

Figure 7 Cracking strains for various values of the vertical loading for the cases of the uncracked wall (left column) and the cracked walls(3rd resolution middle column and 4th resolution right column)

the 5th resolutions of the crack leads to results that are closeenough to those of the uncracked wall There exist somedifferences between the uncracked and the cracked walls forhorizontal displacements in the range of 2mmndash6mm wherethe uncracked wall exhibits greater resistance Howeverwhen the horizontal displacement reaches the value of 6mmthe uncracked wall appears to have strength degradation andafter this displacement value the results of the 4th and the 5thresolutions of the fractal crack are again very close to those ofthe initially uncracked wall

Significantly different is the behaviour of the 3rd res-olution of the fractal crack Although in the first loadingsteps the results are close to those of the 4th and 5thresolutions after a displacement value of 25mm signif-icant strength degradation appears having the form ofsuccessive vertical branches Moreover the ultimate strengthof this wall is significantly lower than the other vari-ants

It is interesting to try to explain this significantly differentbehaviour that appears between the walls corresponding tothe 3rd and the higher resolutions of the fractal crack Forthis reason all the parameters affecting the behaviour of thewall will be comparatively studied in the sequel

Figure 7 depicts the cracking strains of concrete forspecific values of the axial loading All the depicted resultscorrespond to the end of the analysis that is they have beenobtained for an applied horizontal displacement of 20mmFirst of all it can be noticed that for low values of axialloading the cracking patterns that have been developed inall the studied walls are rather similar The larger crackingstrain values (yellow and grey colours) have their nature in thebending deformation of the wall For moderate axial loadingthe cracking patterns are quite different The uncracked wallpresents a bending type cracking pattern The cracked walls(3rd and 4th resolutions) seem to behave differently Both ofthem exhibit significant cracking in the vicinity of the crack


4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN

Figure 8 Plastic strains for various values of the vertical loading for the cases of the uncracked wall (left column) and the cracked walls (3rdresolution middle column and 4th resolution right column)

(grey colours) Moreover shear type cracking patterns aredeveloped at the upper parts of the walls

The above results alone cannot explain the significantlydifferent responses that the two cracked variants of thewall exhibit For this reason the plastic strains of concreteobtained for the same applied horizontal displacement areexamined Figure 8 depicts the plastic concrete strains forthe three different variants of the wall and for specific valuesof axial loading The upper value of the presented scalecorresponds actually to the crushing limit (grey values)Therefore it can be considered that concrete stresses in theseareas are actually zero For the uncracked wall (left columnof Figure 8) it can be noticed that the more heavily deformedregion is the lower right corner It is clear that in this case thewall exhibits a typical bending type deformation behaviour(cracking at the lower left region crushing at the lower rightcorner)

On the other hand the cracked walls seem to deformsignificantly in the vicinity of the crack This phenomenon

is more pronounced in the case of the 3rd resolution of thefractal wall and especially in the case of heavy axial loadingwhere it can be noticed that the vicinity of the crack is inthe crushed state that is the forces are transmitted solelyby the steel mesh in this region For the case of the 4thresolution this phenomenon is rather limited that is it canbe concluded that in this case the crack retains partially itsability to transfer shear and compressive forces through thecontact and friction phenomena developed in the interfaceand through themechanical interlocking that occurs betweenthe two interface parts

In the sequel it is interesting to examine the deformationsthat have occurred at the steel mesh Figure 9 displaysthe steel mesh for the cases studied above The presenteddeformations which correspond to the last load step havebeen magnified by a factor of 10 so that the differencesbetween the examined cases are visible It is obvious thatfor low vertical loading values the deformations of the steelmeshes are actually very similar However for moderate


N = 500kN

N = 1500kN

N = 2500kN

Figure 9 Deformation of the steel mesh for various values of the vertical loading for the cases of the uncracked wall (left column) and thecracked walls (3rd resolution middle column and 4th resolution right column)

values of the vertical loading (119873 = 1500 kN) there existsome differences The steel meshes of the cracked walls seemto be distorted in the vicinity of the right part of the formedcrack In this region the vertical rebars above and belowthe crack present an offset which can be attributed to theinability of the interface to transfer shear forces For the caseof heavy vertical loading the situation is much different Thewall corresponding to the 4th resolution of the fractal crackhas a deformation similar to that of the case of moderate

loading However the steel mesh of the wall correspondingto the 3rd resolution of the fractal crack exhibits significantdeformations along the crack More specifically the uppervertical rebars present a significant horizontal offset withrespect to the lower ones This horizontal offset is obviouseven in the leftmost part of the wall Moreover the horizontalrebars of the upper part present a vertical offset with respectto the ones of the lower part This deformation pattern ofthe steel mesh verifies the findings that were noticed in


2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692

Horizontal displacement 120575 = 5mm


Figure 10 Forces developed in the steel mesh for various values of the vertical loading for the uncracked wall (left column 119873 = 500 kNmiddle column119873 = 1500 kN and right column119873 = 2500 kN)

Figure 8 concerning the excessive strains in the vicinity of thecrack (which had values well above the crushing strain limit)resulting from the inability of concrete to transfer any loadingin this case

Now the difference in the response between the 3rd andthe 4th resolutions of the fractal crack for the case of heavyvertical loading will be explained As it has already beennoticed higher vertical loading leads to higher values of thehorizontal loading The increased horizontal forces have tobe transferred from the upper part of the cracked wall to itslower part In this respect three mechanisms are developedin order to facilitate the horizontal load transfer

(i) exploitation of the tensile strength of the horizontalrebars

(ii) development of friction on the part of the crack wherecontact forces occur

(iii) mechanical interlock between the two interfaces ofthe crack

The first two mechanisms are almost similar in bothcracked walls However it is obvious from Figure 6 thatin higher resolutions fractal crack improves its capacity totransfer forces through the mechanical interlockmechanism

According to the authorsrsquo opinion this fact is the mainreason for the difference in the response between the wallscorresponding to the 3rd and the 4th resolutions of thefractal crack For lower vertical load values the differences arerather limited however as the vertical loading increases theresponse is completely different because the increased verticalforces are combined with the increased horizontal forces andldquodestroyrdquo completely the vicinity of the interface

Figures 10 11 and 12 display the forces developed at thehorizontal and vertical rebars of the steel mesh for the threevariants of the wall examined here for displacements of 5mmand 20mm The left column of each figure corresponds tolower values of the axial loading (119873 = 500 kN) the middlecolumn to moderate loading values (119873 = 1500 kN) and theright column to heavy axial loading (119873 = 2500 kN)

For the case of the uncracked wall (Figure 10) it isnoticed that in the early horizontal loading steps (120575 = 5mm)only the vertical rebars are significantly loaded The rebarsin the left side of the wall have tensile forces while therebars in the right side develop compressive forces as aresult of the bending of the wall For 120575 = 20mm after thedevelopment of cracking in various parts of the initiallyuncracked wall the horizontal rebars are also stressedmainly in the areas where the corresponding cracks have


2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692



Figure 11 Forces developed in the steel mesh for various values of the vertical loading for the 3rd resolution of the crack (left column119873 = 500 kN middle column119873 = 1500 kN and right column119873 = 2500 kN)

reduced or zeroed the ability of concrete to transfer shearforces

For the case of the 3rd resolution of the crack (Figure 11)it is noticed that the vertical rebars are stressed only forsmall axial loading values For moderate and heavy axialloading the vertical rebars are partially only stressed Itdeserves to be noticed that the rebar stresses are negative inthe vicinity of the crack a fact which verifies that the concreteis unable to transfer even compressive loading Moreover itis noticed that the rebars of the right side of the wall donot develop compressive stresses any more due to the factthat the magnitude of bending that develops in this case issignificantly smaller than that in the case of the uncrackedwall The horizontal rebars are stressed only in specific areasnear the crack and in the regions where cracking strains havebeen developed In any case a closer look in the forces thathave been developed in the rebars verifies the significantlydecreased bending capacity of the specific wall

The situation is rather different in the case of the 4thresolution of the fractal crack (Figure 12) It can easily beverified that for small values of the axial loading the pictureof the forces of the vertical rebars is quite similar to that ofthe uncracked wall The same holds also for the forces of thehorizontal rebars For moderate axial load values the forces

of the vertical rebars appear to be discontinuities At theright part of the crack it can be noticed that in some rebarsthe forces are compressive indicating once again the partialinability of concrete in this region to transfer compressiveloading The horizontal rebars are mainly stressed in theupper part of the cracked wall and in the vicinity of the crackThis result is absolutely compatible with the remarks given forthe cracked areas in Figure 7

5 Conclusions

In the paper the finite element analysis of a typical shearwall element which follows the construction practices appliedin Greece during the 70s was presented assuming that acertain crack has been developed as a result of an earthquakeaction The crack was modelled following tools from thetheory of fractals Three different resolutions of the fractalcrack were considered by taking into account the aggregatesizes of the concrete and their results were compared tothose of the initially uncracked wall The main finding ofthe paper is that the cracked wall still has the capacity tosustainmonotonic horizontal loading For small axial loadingvalues this capacity is similar to that of the initially uncrackedwall However for larger axial loading values where the




4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

Figure 12 Forces developed in the steel mesh for various values of the vertical loading for the 4th resolution of the crack (left column119873 = 500 kN middle column119873 = 1500 kN and right column119873 = 2500 kN)

demands increase it seems that a more accurate simulationof the geometry of the fractal crack (ie considering highervalues of the resolution of the interfaces) leads to betterresults Using lower resolution values the roughness ofthe interfaces is not taken into account and therefore themechanical interlock between the two faces of the crack israther limited leading the concrete in the vicinity of thecrack to overstressing and gradually to a complete loss of itscapacity to sustain any kind of forces In this case the bendingcapacity of the wall is significantly limited

References

[1] H-G Sohn Y-M Lim K-H Yun andG-H Kim ldquoMonitoringcrack changes in concrete structuresrdquoComputer-AidedCivil andInfrastructure Engineering vol 20 no 1 pp 52ndash61 2005

[2] T C Hutchinson and Z Chen ldquoImage-based frameworkfor concrete surface crack monitoring and quantificationrdquoAdvances in Civil Engineering vol 2010 Article ID 215295 2010

[3] B B Mandelbrot D E Passoja and A J Paullay ldquoFractalcharacter of fracture surfaces of metalsrdquo Nature vol 308 no5961 pp 721ndash722 1984

[4] V E Saouma C C Barton and N A Gamaleldin ldquoFractalcharacterization of fracture surfaces in concreterdquo EngineeringFracture Mechanics vol 35 no 1ndash3 pp 47ndash53 1990

[5] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquoWear vol 136 no 2 pp 313ndash3271990

[6] V E Saouma and C C Barton ldquoFractals fractures and sizeeffects in concreterdquo Journal of Engineering Mechanics vol 120no 4 pp 835ndash854 1994

[7] A Carpinteri and B Chiaia ldquoMultifractal nature of concretefracture surfaces and size effects on nominal fracture energyrdquoMaterials and Structures vol 28 no 8 pp 435ndash443 1995

[8] B Chiaia J G M Van Mier and A Vervuurt ldquoCrack growthmechanisms in four different concretes microscopic observa-tions and fractal analysisrdquo Cement and Concrete Research vol28 no 1 pp 103ndash114 1998

[9] H Mihashi N Namura and T Umeoka ldquoFracture mechanicsparameters of cementitious composite materials and fracturedsurface propertiesrdquo in Europe-US Workshop on Fracture andDamage in Quasibrittle Structures Prague Czech Republic1994

[10] A Carpinteri and P Cornetti ldquoSize effects on concrete tensilefracture properties an inter-pretation of the fractal approachbased on the aggregate gradingrdquo Journal of the MechanicalBehavior of Biomedical Materials vol 13 no 3-4 pp 233ndash2462011

[11] H-Q Sun J Ding J Guo and D-L Fu ldquoFractal research oncracks of reinforced concrete beams with different aggregates


sizesrdquoAdvancedMaterials Research vol 250-253 pp 1818ndash18222011

[12] V Mechtcherine ldquoFracture mechanical behavior of concreteand the condition of its fracture surfacerdquo Cement and ConcreteResearch vol 39 no 7 pp 620ndash628 2009

[13] M Borri-Brunetto A Carpinteri and B Chiaia ldquoScaling phe-nomena due to fractal contact in concrete and rock fracturesrdquoInternational Journal of Fracture vol 95 no 1ndash4 pp 221ndash2381999

[14] O Panagouli and E Mistakidis ldquoDependence of contact areaon the resolution of fractal interfaces in elastic and inelasticproblemsrdquo Engineering Computations vol 28 no 6 pp 717ndash746 2011

[15] E Mistakidis and G Stavroulakis Nonconvex Optimization inMechanics Algorithms Heuristic and Engineering Applicationsby the FEM Kluwer 1997

[16] A Carpinteri B Chiaia and S Invernizzi ldquoThree-dimensionalfractal analysis of concrete fracture at the meso-levelrdquoTheoret-ical and Applied Fracture Mechanics vol 31 no 3 pp 163ndash1721999

[17] GMourot SMorel E Bouchaud andGValentin ldquoAnomalousscaling of mortar fracture surfacesrdquo Physical Review E vol 71no 1 Article ID 016136 2005

[18] A Carpinteri G Lacidogna and G Niccolini ldquoFractal analysisof damage detected in concrete structural elements underloadingrdquo Chaos Solitons amp Fractals vol 42 no 4 pp 2047ndash2056 2009

[19] BMandelbrotTheFractal Geometry ofNatureWH Freemann1982

[20] B B Mandelbrot ldquoSelf-affine fractals and fractal dimensionrdquoPhysica Scripta vol 32 no 4 pp 257ndash260 1985

[21] K J Maloslashy A Hansen E L Hinrichsen and S Roux ldquoExper-imental measurements of the roughness of brittle cracksrdquoPhysical Review Letters vol 68 no 2 pp 213ndash215 1992

[22] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of surfacesrdquoJournal of Tribology vol 112 no 2 pp 205ndash216 1990

[23] P D Panagiotopoulos and O K Panagouli ldquoFractal geometryin contact mechanics and nu-merical applicationsrdquo in CISM-Book on Scaling Frac-Tals and Fractional Calculus in ContinumMechanics A Carpintieri and F Mainardi Eds pp 109ndash171Springer 1997

[24] F M Borodich and D A Onishchenko ldquoSimilarity and frac-tality in the modelling of roughness by a multilevel profilewith hierarchical structurerdquo International Journal of Solids andStructures vol 36 no 17 pp 2585ndash2612 1999

[25] FM Borodich andA BMosolov ldquoFractal roughness in contactproblemsrdquo Journal of Applied Mathematics and Mechanics vol56 no 5 pp 681ndash690 1992

[26] E S Mistakidis and O K Panagouli ldquoStrength evaluation ofretrofit shear wall elements with interfaces of fractal geometryrdquoEngineering Structures vol 24 no 5 pp 649ndash659 2002

[27] E S Mistakidis and O K Panagouli ldquoFriction evolution as aresult of roughness in fractal interfacesrdquo Engineering Computa-tions vol 20 no 1-2 pp 40ndash57 2003

[28] C-J Chen T-Y Lee Y M Huang and F-J Lai ldquoExtraction ofcharacteristic points and its fractal reconstruction for terrainprofile datardquo Chaos Solitons amp Fractals vol 39 no 4 pp 1732ndash1743 2009

[29] M Barnsley Fractals Everywhere Academic Press 1988

[30] G-D Hu P D Panagiotopoulos P Panagouli O Scherf and PWriggers ldquoAdaptive finite element analysis of fractal interfacesin contact problemsrdquo Computer Methods in Applied Mechanicsand Engineering vol 182 no 1-2 pp 17ndash37 2000

[31] R de Borst J J C Remmers A Needleman andM-A AbellanldquoDiscrete vs smeared crack models for concrete fracture bridg-ing the gaprdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 28 no 7-8 pp 583ndash607 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


contact regions are strongly influenced by the roughness ofthe contacting surfaces

The multiscale nature of the surface roughness suggeststhe use of fractal geometryThe fractal approach adopted herefor the simulation of the geometry of the cracks formed ina shear wall uses computer generated self-affine curves forthe modelling of the interface roughness which is stronglydependent on the values of the structural parameters ofthese curves The computer generated fractal interfaceswhich are based on a given discrete set of interface dataare characterized by a precise value of the resolution 120575 ofthe interface This fact permits the study of the interfaceroughness on iteratively generated rough profiles makingthis approach suitable for engineering problems since itpermits the satisfactory study of the whole problem withreliable numerical calculations

The aim of this paper is to study how the resolutionof a fractal crack F affects the strength of a reinforcedconcrete shear wall element On the interface between thetwo cracked surfaces unilateral contact and friction con-ditions are assumed to hold The applied approach takesinto account the nonlinear behaviour of the materialsincluding the limited strength of concrete under tensionThe shear wall is submitted to shear loading As a resultof the applied approach the contribution of the frictionbetween the cracked surfaces is taken into account as wellas the additional strength coming from the mechanicalinterlock between the two faces of the crack For everystructure resulting for each resolution of the interface aclassical Euclidean problem is solved by using a variationalformulation [15] It must be mentioned here that the finestresolution of the interface is related to the size of theaggregates

2 Fractal Representation of Rough Surfaces

The fractal nature of material damage has been a matter ofa very intense research during the last three decades Thefractal nature of fractured surfaces in metals was shownmore than 30 years ago by Mandelbrot et al [3] Morespecifically it was shown that the fractured surfaces inmetals develop a fractal structure over more than threeorders of magnitude In quasibrittle materials observationshave shown that fractured surfaces display self-affine scaleproperties in a certain range of scales which is in mostcases very large and which greatly depends on the materialmicrostructure This is true for a large variety of quasibrittlematerials such as rock concrete and ceramics [4ndash8 16ndash18]

Fractal sets are characterized by noninteger dimensions[19] The dimension of a fractal set in plane can vary from0 to 2 Accordingly by increasing the resolution of a fractalset its length tends to 0 if its dimension is smaller than 1(totally disconnected set) or tends to infinity if it is largerthan 1 In these cases the length is a nominal useless quantitysince it changes as the resolution increases Converselythe fractal dimension of a fractal set is a parameter of

great importance because of its scale-independent charac-ter

Many methods which are based on experimental ornumerical calculations such as the Richardson method[19] have been developed for the estimation of the fractaldimension of a curve According to this method dividerswhich are set to a prescribed opening120575 are usedMovingwiththese dividers along the curve so that each new step startswhere the previous step leaves off one obtains the numberof steps 119873(120575) The curve is said to be of fractal nature if byrepeating this procedure for different values of 120575 the relation

119873(120575) sim 120575minus119863 (1)

is obtained in some interval 120575lowast lt 120575 lt Δlowast The power 119863denotes the fractal dimension of the profile which is in therange 1 lt 119863 lt 2 The relation between the fractal dimension119863 of this profile and the dimension of the correspondingsurface is119863

119904

= 119863 + 1 [16]In relation (1) there is an upper and a lower bound

in the scaling range and consequently a transition fromthe fractal regime at the microscopic level to the Euclideanregime at largest scales The upper bound is representedby the macroscopic size of the set while the lower oneis related to the size of the smallest measurable particlesthat is the aggregates in the case of concrete Mandelbrot[20] first pointed out the transition from a fractal regimecharacterized by noninteger dimensions to the homogeneousone characterized by classical topological dimensions a factwhich points out the main difference between mathematicaland physical fractals

The idea of self-affinity is very popular in studying surfaceroughness because experimental studies have shown thatunder repeated magnifications the profiles of real surfacesare usually statistically self-affine to themselves [3 21] Theself-affine fractals were used in a number of papers as atool for the description of rough surfaces [22ndash28] Typicallysuch a profile can be measured by taking height data 119910

119894

with respect to an arbitrary datum at 119873 equidistant discretepoints 119909

119894

In the sequence fractal interpolation functionsF(119909119894

) = 119910119894

119894 = 0 1 119873 are used for the passage fromthis discrete set of data (119909

119894

119910119894

) 119894 = 0 1 2 119873 to acontinuous model According to the theory of Barnsley [29]the sequence of functionsF

119899+1

(119909) = (119879F119899

)(119909) = 119888119894

119897119894

minus1

(119909) +

119889119894

F119899

(119897119894

minus1

(119909)) + 119892119894

converges to a fractal curveF as 119899 rarr infinThe transformation 119897

119894

transforms [1199090

119909119873

] to [119909119894minus1

119909119894

] and itis defined by the relation 119897

119894

(119909) = 119886119894

119909 + 119887119894

The calculation ofthe parameters 119886

119894

119887119894

119888119894

and119892119894

is based on the given set of dataand the free parameters 119889

119894

Fractal interpolation functions give profiles which look

quite attractive from the viewpoint of a graphic roughnesssimulation In higher approximations these profiles appearrougher as it is shown in the next section where the first tofifth approximations of a fractal interpolation function arepresented Moreover the roughness of the profile is stronglyaffected by the free parameters 119889

119894

of the interpolation func-tions As these parameters take larger values the resultingprofiles appear rougher with sharper peaks


qN

P 120575

Fixed boundary

1500

2950

y

x 500

4000

4500




119899



119873

sum

119894=1

10038161003816100381610038161198891198941003816100381610038161003816 119886119863minus1

119894

= 1 (2)






(m)1st 1404 48882nd 0468 49463rd 0156 50804th 0052 53735th 0017 5935


1

=

1198892

= 1198893



119899 = 1 2


119899


119899


119899



119873






5th resolution




=




0004

136

16

Compression

Tension

120590 (MPa)

ks120576

(a)

220

02

264

minus220minus264

120590 (MPa)

120576ay

(b)

400

02

480

minus400minus480

120590 (MPa)

120576ay

(c)






5










4 Numerical Results






0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

Uncracked wall

0

200

400

600

800

1000

1200

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


N = 0

N = 500

N = 1000

N = 1500

N = 2000

N = 2500


0

100

200

300

400

500

600

700

800

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

N = 0

0

200

400

600

800

1000

1200

1400

0 0005 001 0015 002

Forc

e (kN

)Fo

rce (

kN)

Displacement (m)

N = 1500

0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002Displacement (m)

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN








N = 500kN

N = 1500kN

N = 2500kN





2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692














2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692








5 Conclusions





4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000



References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










qN

P 120575

Fixed boundary

1500

2950

y

x 500

4000

4500




119899



119873

sum

119894=1

10038161003816100381610038161198891198941003816100381610038161003816 119886119863minus1

119894

= 1 (2)






(m)1st 1404 48882nd 0468 49463rd 0156 50804th 0052 53735th 0017 5935


1

=

1198892

= 1198893



119899 = 1 2


119899


119899


119899



119873






5th resolution




=




0004

136

16

Compression

Tension

120590 (MPa)

ks120576

(a)

220

02

264

minus220minus264

120590 (MPa)

120576ay

(b)

400

02

480

minus400minus480

120590 (MPa)

120576ay

(c)






5










4 Numerical Results






0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

Uncracked wall

0

200

400

600

800

1000

1200

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


N = 0

N = 500

N = 1000

N = 1500

N = 2000

N = 2500


0

100

200

300

400

500

600

700

800

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

N = 0

0

200

400

600

800

1000

1200

1400

0 0005 001 0015 002

Forc

e (kN

)Fo

rce (

kN)

Displacement (m)

N = 1500

0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002Displacement (m)

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN








N = 500kN

N = 1500kN

N = 2500kN





2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692














2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692








5 Conclusions





4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000



References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












5th resolution




=




0004

136

16

Compression

Tension

120590 (MPa)

ks120576

(a)

220

02

264

minus220minus264

120590 (MPa)

120576ay

(b)

400

02

480

minus400minus480

120590 (MPa)

120576ay

(c)






5










4 Numerical Results






0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

Uncracked wall

0

200

400

600

800

1000

1200

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


N = 0

N = 500

N = 1000

N = 1500

N = 2000

N = 2500


0

100

200

300

400

500

600

700

800

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

N = 0

0

200

400

600

800

1000

1200

1400

0 0005 001 0015 002

Forc

e (kN

)Fo

rce (

kN)

Displacement (m)

N = 1500

0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002Displacement (m)

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN








N = 500kN

N = 1500kN

N = 2500kN





2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692














2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692








5 Conclusions





4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000



References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0004

136

16

Compression

Tension

120590 (MPa)

ks120576

(a)

220

02

264

minus220minus264

120590 (MPa)

120576ay

(b)

400

02

480

minus400minus480

120590 (MPa)

120576ay

(c)






5










4 Numerical Results






0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

Uncracked wall

0

200

400

600

800

1000

1200

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


N = 0

N = 500

N = 1000

N = 1500

N = 2000

N = 2500


0

100

200

300

400

500

600

700

800

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

N = 0

0

200

400

600

800

1000

1200

1400

0 0005 001 0015 002

Forc

e (kN

)Fo

rce (

kN)

Displacement (m)

N = 1500

0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002Displacement (m)

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN








N = 500kN

N = 1500kN

N = 2500kN





2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692














2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692








5 Conclusions





4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000



References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












4 Numerical Results






0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

Uncracked wall

0

200

400

600

800

1000

1200

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


N = 0

N = 500

N = 1000

N = 1500

N = 2000

N = 2500


0

100

200

300

400

500

600

700

800

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

N = 0

0

200

400

600

800

1000

1200

1400

0 0005 001 0015 002

Forc

e (kN

)Fo

rce (

kN)

Displacement (m)

N = 1500

0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002Displacement (m)

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN








N = 500kN

N = 1500kN

N = 2500kN





2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692














2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692








5 Conclusions





4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000



References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

Uncracked wall

0

200

400

600

800

1000

1200

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)


N = 0

N = 500

N = 1000

N = 1500

N = 2000

N = 2500


0

100

200

300

400

500

600

700

800

0 0005 001 0015 002

Forc

e (kN

)

Displacement (m)

N = 0

0

200

400

600

800

1000

1200

1400

0 0005 001 0015 002

Forc

e (kN

)Fo

rce (

kN)

Displacement (m)

N = 1500

0

200

400

600

800

1000

1200

1400

1600

0 0005 001 0015 002Displacement (m)

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN








N = 500kN

N = 1500kN

N = 2500kN





2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692














2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692








5 Conclusions





4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000



References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN







4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN








N = 500kN

N = 1500kN

N = 2500kN





2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692














2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692








5 Conclusions





4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000



References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000

N = 500kN

N = 1500kN

N = 2500kN








N = 500kN

N = 1500kN

N = 2500kN





2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692














2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692








5 Conclusions





4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000



References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










N = 500kN

N = 1500kN

N = 2500kN





2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692














2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692








5 Conclusions





4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000



References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692














2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692








5 Conclusions





4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000



References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2640

2107

1574

1040

507

minus026

minus559

minus1092

minus1625

minus2159

minus2692








5 Conclusions





4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000



References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












4000e minus 003

3600e minus 003

2800e minus 003

3200e minus 003

2400e minus 003

2000e minus 003

1600e minus 003

1200e minus 003

8000e minus 004

4000e minus 004

0000e + 000



References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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