Cracking Patterns of Shear Walls in Reinforced Concrete Structure due to Strong Earthquake Based on Mohr－Coulomb Criterion

ZHA Zhi－xiang(查支祥)，LIU Xi－la(劉西拉)，PENGWei(彭衛(wèi))，XU Bo(許波)

1 School of Naval Architecture，Ocean and Civil Engineering，Shanghai Jiao Tong University，Shanghai200240，China

2 School of Civil Engineering and Architecture Ningbo Institute of Technology，Zhejiang University，Ningbo 315100，China

3 School of Civil Engineering，Henan Polytechnic University，Jiaozuo 454003，China

Cracking Patterns of Shear Walls in Reinforced Concrete Structure due to Strong Earthquake Based on Mohr－Coulomb Criterion

ZHA Zhi－xiang(查支祥)1，2，LIU Xi－la(劉西拉)1*，PENGWei(彭衛(wèi))2，XU Bo(許波)1，3

1 School of Naval Architecture，Ocean and Civil Engineering，Shanghai Jiao Tong University，Shanghai200240，China

2 School of Civil Engineering and Architecture Ningbo Institute of Technology，Zhejiang University，Ningbo 315100，China

3 School of Civil Engineering，Henan Polytechnic University，Jiaozuo 454003，China

A primary goal of seism ic design of buildings is to protect peop le's life safety during strong earthquakes．Fundamentally，to predict the survived space and estimate people escape intervals during structural collapse are very important，which requires to describe the failure process more detailed ly not only for structural joints but also for slabs and shear walls．In the present paper，the details of M ohr-Coulomb failure criterion with tension-cutoff together with its expression in failure surface and local coordinate system of reinforced concrete(RC)shear wall are given firstly，and then several typical cracking patterns of RC shear wall，such as tension shear crack and compression shear crack，are analyzed based on M ohr-Coulomb failure criterion with tension-cutoff．

cracking patterns;Mohr－Coulomb criterion;shear wall

Introduction

Reinforced concrete(RC)frame－wall combined by RC frame and RC shear wall is one of the main structure types of the high－rise buildings.In fact，RC shear wall panels functionally resist lateral loads acting on the high－rise buildings. Post earthquake studies show that buildings with shear walls resist the seism ic forces efficiently and generally do not collapse undermoderate to severe earthquake［1］.

In recent years，substantial research has been carried outon seismic response of buildings with RC frame－wall［2－7］，but the collapse analysis of RC frame－wall buildings under the strong earthquakes is very little.

Research on the collapsemechanism of RC frame structures under disastrous loading including strong earthquake has been well finished［8］.In fact，the RC wall panel with the different damage degree plays a different role on the frame－wall structure during the strong earthquakes.Research on the collapse of RC wall panel combined with the known RC frame collapse analysis is the key on the collapse analysis of RC frame－wall buildings.

Cracking failure of concrete is the onset of the wall panel collapse，the process experienced crack progression，steel bar yielding，reinforcement fracture，and concrete crush.In the present paper，the details of Mohr－Coulomb failure criterion with tension－cutoff togetherwith its expression in failure surface and local coordinate system of RC shear wall are given firstly and then several typical cracking patterns of RC shear wall，such as tension shear crack and compression shear crack，have been analyzed based on Mohr－Coulomb failure criterion with tension－cutoff，which can be used for further collapse analysis of RC frame－wall structures.

1 Two Kinds of Coordinate System s of RC Shear Wall

For the sake of convenience，two kinds of coordinate systems of shear wall including failure surface varied with stress state and fixed local coordinate x－y systems of shear wall are established，as shown in Fig.1.A crack is assumed to form in the x'－y'plane named failure surfaces if stress fracture criteria are reached.In order to avoid the complexities of the problem，it is assumed that square wall panel is loaded with invariable vertical load N and increasing horizontal load F.

Fig.1 Relationship between normal stress and shear stress on the structural failure surface in local coordinate system

2 Mohr-Coulomb's Criterion with Tension-Cutoff

Many proposals of failure criteria for concrete can be found in Ref.［9］.The most commonly used failure criteria are defined in stress spaces by a number of material constants varying from one to five independent control parameters.The brittle fracture of concrete under tensile and small compressive stresses is best described by themaximum tensile stress criterion of Rankine.For concrete in the high hydrostatic pressure range，shearing stress criteria，such as Tresca and von－M ises criteria，are used to predict the ductile fracture.For a pressure－dependent material，the deviatoric cross sections of the failure envelope in the principal stress space along the hydrostatic axis are different in size and need not to be geometrically similar in general.The simplest and most commonly used models of this type are the Mohr－Coulomb and Drucker－Prager failure criteria.These simpler models have been refined and generalized by adding additional parameters，and various three－，four－，and fiveparametermodels have been formulated.

For a long time，failure of concrete under general stress states has been based on the Mohr－Coulomb failure criterion combined with tension－cutoff(themaximum tensile stress).Inmany cases，this combined criterion provides a fair first approximation for concrete failure.

According to this combined criterion，concrete failure can be divided into two types:tensile(or cracking)type and compressive(or shear slip)type.In the tension－cutoff range，the tensile type of cleavage fracture occurs because of splitting or cracking，and it is governed by the criterion of a constant maximum tensile stress.In the compressive stress range，the criterion states that compressive type of failure occursbecause of shear slips，and it is governed by themaximum shear stress of Mohr－Coulomb type.The failure envelope of concrete composed of failure points(σ，)at structural failure surface is shown in Fig.2.Itshould be noted thatwhenσ＞0 itmeans the normal stress at the failure point is compressive and“compression shear failure”happens.

Fig.2 Mohr－Coulomb criterion with tension－cutoff(critical stress state)

2．1 M ohr-Coulomb's criterion with tension-cutoff at structural failure surface

In 1992，Li et al．presented a critical tensile－shear failure state of plain concrete and corresponding critical stress based on the concrete micro crack propagation model［10］.A significant Mohr's stress circle of critical stress is tangent to the lines of Mohr－Coulomb shear failure criterion and Rankine tensile failure criterion simultaneously.The tangency point of Mohr's stress circle of critical stress and the line of Mohr－Coulomb shear failure criterion are situated in(o，c)，as shown in Fig.2.At this Mohr's stress circle corresponding to the critical stress，the points of intersection withσ－axis are defined asσ1candσ3c，respectively.The coordinates ofσ1candσ3ccan be found as follows.

From Fig.2，

From Eqs.(1)and(2)

In Fig.2，σ3ccan be expressed as

From Eqs.(1)，(2)，and(4)

For the sake of finding the relationship of three parameters c，ft，and fc，the uniaxial compressive failure test is studied，shown in Fig.3.

Fig.3 Relationship between normal stress and shear stress on the structural failure surface under uniaxial compressive stress statewith Mohr－Coulomb criterion with tension－cutoff

From the uniaxial compressive test，

From Eq.(6)，

Comparing Eq.(5)with Eq.(7)，in the critical failure state，σ3ccan be expressed as

The equation of Mohr circles of critical tensile－shear failure stress state，shown in Fig.2 is

Substituting the stress condition A(0，c)in Fig.2 into the Eq. (9)，σ3ccan be found as

From Eqs.(8)and(10)it can be found that

2．2 M ohr-Coulomb's criterion with tension-cutoff in local coordinate system

The points B(σy，τyx)and C(0，τxy)in Fig.4 represent two stress states of two－perpendicular planes，which are socalled stress states in the local coordinate system x－y，as shown in Fig.1.

The equations of Mohr stress circle in Mohr－Coulomb failure state(shown in Fig.4)are

Fig.4 Mohr－Coulomb criterion with tension－cutoff in general condition

From the Mohr－Coulomb criterion in the compressive stress range，we get

It can be obtained from Eqs.(14)－(16)that

From Eq.(17)，it can be expressed as

From Eq.(7)，

For the Rankine criterion of concrete，as shown in Fig.2，the concrete will failwhenσ=ftandτ=0．

From Eq.(12)，the equation of Mohr stress circle in Rankine failure state is expressed，

2．3 Critical stress state of M ohr-Coulomb's criterion with tension-cutoff in local coordinate system

As shown in Fig.2，the Mohr circle represents a critical stress state A(0，c)，where Mohr－Coulomb shear failure and tension failure occur at the same time.By the relationship between normal stress and shear stress on the structural failure surface in local coordinate system，as shown in Fig.1，the critical stresses of point A on the structural failure surface can be expressed in local coordinate system.

The point B coordinate valueswith the corresponding point A(0，c)can be obtained from Figs.1 and 2 that

Thus，it can be obtained from Eqs.(11)，(19)－(21) that

Mohr－Coulomb's criterion with tension－cutoff in local coordinate systemis shown in Fig.5 from Eqs.(21)， (23)，and(24).In Fig.5，the coordinate value of the critical stress state point M'of tension shear failure and compression shear failure on structural failure surface in local coordinate system can be shown as0．5+ft/fcandIt can be proved that the intersection point(0．5+ft/fc，of Mohr－Coulomb criterion and Rankine criterion represents not only the critical stress state，butalso the stationary pointof Mohr－Coulomb's criterion with tension－cutoff in Fig.5.

Fig.5 Mohr－Coulomb criterion with tension－cutoff in the local coordinate system x－y

3 Patterns of Cracks in RC ShearWalls

Concrete cracking，buckling/yield and rupture of reinforcement are the most important factors giving rise to the strength and stiffness deterioration and even collapse for RC shear walls.In the present paper，cracking patterns and propagation of RC shear wall will be discussed in detail on the basis of Mohr－Coulomb criterion with tension－cutoff in the local coordinate system as justmentioned above.

The follow ing basic assumptions can bemade to distinguish cracking patterns of concrete shear wall.

(1)For the frame－wall system，the shear stress taken by frame column is little，which can be neglected.

(2)Shear stress in thehorizontal section of web concrete plate of frame－wall is distributed uniform ly.

3．1 Typical crack patterns

Basically，the crack patterns for concrete shearwalls can be classified into two types，namely tension shear crack and compression shear crack，as shown in Fig.5.

The typical real crack patterns of RC shear wall are shown in Fig.6［2－7，11］.

Fig.6 Typical crack patterns of RC shear walls

3．2 Crack criterion

Asmentioned above，the Mohr－Coulomb's criterion with tension－cutoff can provide a fair first approximation for the fracture of concrete after macro cracking.But for a given concrete wall with ratios of height to w idth and ratios of compressive stress to concrete strength under vertical and horizontal loadings，as shown in Fig.7，thewallmaybe crack at some area such as BR，BL，DL，and DR．How to judge which kind of crack will occur first?The crack criterion of concrete will be presented as follows.Only web plate of RC shearwall is discussed here.

Fig.7 Typical crack patterns of RC shear walls

1)Which crack will occur first in BRarea and BLarea?

As shown in Fig.8 the line MM'divides the area enclosed by the Mohr－Coulomb's criterion with tension－cutoff and abscissa axis into two parts，partⅠand partⅡ，namely the tension shear area and the compression shear area，respectively.

The point(σy/fc，τyx/fc)located in the area AA1A2represents the normal stress，summed byσNandσM，and the shear stress induced by horizontal force at the top ofwall.Here，The normal stressσ(pointNA in Fig.8)，assumed as a constant in the present paper，which is the uniform distribution along the section height of wall is induced by the vertical load N.The normal stressσMwhich is the liner distribution along the sectional height of wall is also induced by the horizontal force.

Obviously，

where F is the applied shear force，M is theweb sectionmoment of the shear wall，W is the web section modulus，h is the web section height，b is the web section w idth，and H is the web height of shear wall.

Fig.8 Mohr－Coulomb's criterion with tension－cutoff in the local coordinate system

That is to say，the normal stressσMof two edges of shear wallweb increases linearly with the shear stressτyx，as the lines AA1and AA2shown in Fig.8，until to arrive at the failure curve. For a given configuration concretewall，the judgmentwhatkind of crack will first occur under the continuously increased horizontal force is definite.As shown in Fig.8，because the normal stressσNinduced by the vertical load N is assumed as a constant and the shear stress in all cross section of shear wall induced by the increased horizontal force F is always the same，the tension shear crack firstly occurs if the less shear stress betweenτA1andτA2is located at the failure envelope in partⅠ. Inversely，the compression shear crack occurs if the less shear stress betweenτA1andτA2is located at the failure envelope in partⅡ.Obviously，the normal stress valueσNatpoint A and the slopeτyx/σMof lines AA1and AA2are important influence factors deciding crack type.TheσNis related to the axial force and cross section size of shear wall andτyx/σMis related to the height，cross section size，and shape of shear wall.

Now，for a given shearwall configuration，itshould exista critical normal stressσNinduced by vertical force，as point B shown in Fig.9，where the tension shear crack and the compression shear crack occur at the same time，that is to say，the lines BB1and BB2arrive to the failure curve simultaneously and the shear stresses of points B1and B2are the same.The tension shear crack occurs first when the real normal stress induced only by vertical load decreases to point A under the same configuration of wall，similarly，the compression shear crack occurs first when the real normal stress induced only by vertical load increases to point C.

Fig.9 Mohr－Coulomb's criterion with tension－cutoff in the local coordinate system

For any given shear stressτyx/fc，there exist two normal stressesσB1andσB2when the two kinds of cracks occur at the same time，as follows

Furthermore，

and the absolutely maximum value of normal stress induced by the horizontal force，

Obviously，

where H0is the criticalwall heightwith tension shear crack and compression shear crack occurring at the two edges of shearwall web simultaneously，N0is the critical axial force with tension shear crack and compression shear crack occurring at the two edges of shear wall web simultaneously，and A is the web section area.

For any given shear stressτyxacting on shear wall with rectangular web section，there always exists a unique combination(H0/h，N0/(fcA))．The relationship between H0/h and N0/fcA can be figured out by the curves in Fig.10.

Fig.10 Relationship between H0/h and N0/(fcA)

When the configuration of shear wall is given，a critical axial compression ratio N0/(fcA)according to the given heightw idth ratio H0/h can be obtained by the above relationshipcurve.

Whenμ=N0/(fcA)，the tension shear crack in BLarea and the compression shear crack in BRarea，as shown in Fig.7，occur simultaneously;whenμ＞N0/(fcA)，compression shear crack in BRarea occurs firstly;whenμ＜N0/(fcA)，tension shear crack in BLarea occurs firstly，whereμcan be called as the real axial compression ratio.

It is noteworthy that the critical axial compression ratio N0/(fcA)trends to a constant value，being independent of the real height－w idth ratio H0/h when the real height－w idth ratio exceeds about2.

2)Which crack will occur first in BLarea and DLarea in Fig.7?

The emergence of the tension shear crack or com pression shear crack depends on the normal stresswhich is composed ofσNinduced from vertical load andσHfrom horizontal load.

Fig.11 Mohr－Coulomb's criterion with tension－cutoff in the local coordinate system(σN＞σlim)

TheσNandσlimcan be calculated by the given vertical load N and parameters fc，ft，and A.From Eq.(24)，σlim= 0．5fc+ft．

(1)σN≥σlim

Figure 11 indicates Mohr－Coulomb's criterion with tensioncutoff in the local coordinate system.TheσNrepresents the normal stress in web concrete plate section of wall induced by vertical load.Theσlimrepresents the critical normal stress in web concrete plate section of wall.

The process of establishing criterion which will occur first between tension shear crack and compression shear crack is shown as follows.

①Calculation ofσN

②Calculation ofτ0

③Calculation ofσH

BecauseσNis greater thanσlim，there must exist a corresponding normal stressσH，which has the same shear stress τ0with the normal stressσNif the tension shear failure and compression shear failure occur at the same time.The tension shear failure willoccur first if the value of normal stressσis less than the value of critical normal stressσH，sim ilarly，the compression shear failure also will occur first if the value of normal stressσismore than the value of critical normal stress σH．

④Calculation of the criticalwall height H0

On the top of the wall plate shown in Fig.7，

From Eqs.(35)and(36)

From Eqs.(34)－(35)and(37)

Figure 11 shows that the tension shear crack will occur first，if the inequalityσ＜σHis tenable，namely the real ratio of height and w idth of wall ismore than the critical ratio of height and w idth of wall，Sim ilarly，the compression shear crack will occur first，if the inequalityσ＞σHis tenable，that is the inequalityis tenable.The normal stress of wall，tension strength and compression strength of concrete are the important influence factors of the critical ratio of height and w idth of wall.

(2)σN＜σlim

Figure12 indicates the tension shear cracking alwaysoccurs firstly when the inequalityσN＜σlimis tenable，which is independent of the height－width ratios of wall because of their different shear stresses.

Fig.12 Mohr－Coulomb's criterion with tension－cutoff in the local coordinate system(σN＜σlim)

3)Which crack will occur first in BRarea and DRarea in Fig.7?

(1)σN＜σlim

Figure 13 shows that the tension shear crack in the upper part of wallwill occur first，if the inequalityσ＜σHis tenable，that is the real ratio of heightand w idth of wall ismore than the critical ratio of height and w idth of wall，Similarly， the compression shear crack in the lower part of wallwill occur first，if the inequalityσ＞σHis tenable，that is，the inequalityis tenable.

Fig.13 Mohr－Coulomb's criterion with tension－cutoff in the local coordinate system(σN＜σlim)

(2)σN≥σlim

Figure 14 indicates the compression shear crack in the lower part of wall always occurs firstly when the inequality σN≥σlimis tenable，which is independent of the height－w idth ratios of wall.

Fig.14 Mohr－Coulomb's criterion with tension－cutoff in the local coordinate system(σN＞σlim)

4 Conclusions

(1)The expression of Mohr－Coulomb's criterion with tension－cutoff can be converted in different coordinate systems，such as at the failure space and in the local coordinate system，and it can be applied to forming the cracking criterion of RC shear wall conveniently.

(2)Tension shear crack and compression shear crack are the two basic crack patterns of concrete wall panel.Heightw idth ratio of wall and axial compression ratio are the important factors of deciding crack pattern.

(3)In general case，the critical axial compression ratio N0/(fcA)trends to a fixed value，being independentof the real height－w idth ratio H0/h when the real height－width ratio of wall panel exceeds about2.

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TU973

A

1672－5220(2015)04－0541－08

date:2014－02－27

s:Ningbo Scientific and Technological Innovation Team，China(No.2011B81005);Zhejiang Communication Department Foundation，China(No.2013H54);Ningbo Communication and Transportation Comm ittee Foundation，China(No.201314);Zhejiang Construction Bureau Project，China(No.2014Z108)

*Correspondence should be addressed to LIU Xi－la，E－mail:xilaliu@tsinghua.edu.cn