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Multi-directional modeling of crack pattern in 2D R/C members

R. Cerioni *, I. Iori, E. Michelini, P. Bernardi

Department of Civil and Environmental Engineering and Architecture, University of Parma, Parma 43100, Italy

Received 21 December 2006; received in revised form 3 April 2007Available online 29 April 2007

Abstract

A macro-scale approach to R/C modeling is proposed in this paper by formulating a comprehensive model, thatdescribes R/C behavior in the uncracked stage (solid concrete) and in the cracked stage, the latter with either unidirec-tional cracking (primary cracks) or with bidirectional cracking (primary and secondary cracks) or even multi-directionalcracking. The secant stiffness matrix is formulated by means of a direct procedure, based on the assumption that the solidconcrete and the reinforcement work in parallel, while the solid concrete between the cracks and the cracks themselveswork in series. The resistant mechanisms active at the crack interface are introduced by means of their highly nonlinearlaws, that are taken from the literature and are based on well-documented tests. The reliability and accuracy of theproposed model are checked against a few well-documented tests on 2D R/C members failing past the formation ofsecondary cracks. 2007 Elsevier Ltd. All rights reserved.

Keywords: Reinforced concrete; Primary crack; Secondary crack; Concretesteel interaction; Crack surface interaction; Nonlinearanalysis

1. Introduction

In reinforced concrete (R/C) members subjected to plane stresses, mainly loaded in tension and shear, theprediction of crack pattern and its evolution as loading increases is a very complex problem [16]. When thefirst (primary) cracks form, crack pattern shows few cracks with orientation depending on stress field and onspacing and arrangement of reinforcing bars. Normal and shear stresses are transferred across cracks through

complex phenomena, as aggregate interlock and confinement actions, aggregate bridging effect, dowel action,tension stiffening and kinking effects of steel bars, etc. Material discontinuities due to cracks cause a markedchange of stress and strain fields in concrete and in reinforcing steel respect to that observed in the correspond-ing uncracked phase. Therefore, as loading increases, new (secondary) cracks, oriented along different direc-tions with respect to primary cracks, with a smaller spacing, can form. The re-orientation of cracks is strictlyconnected to the anisotropic behaviour of R/C elements, above all when a markedly different steel ratio isadopted for reinforcing steel bar layer.

0013-7944/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfracmech.2007.04.012

* Corresponding author. Tel.: +39 0521 905928; fax: +39 0521 905924.E-mail address: [email protected] (R. Cerioni).

Available online at www.sciencedirect.com

Engineering Fracture Mechanics 75 (2008) 615628

www.elsevier.com/locate/engfracmech
mailto:[email protected]:[email protected]

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In order to formulate an effective model which is able to describe the nonlinear response up to failure ofR/C members, the actual behavior of concrete and steel as well as the interaction phenomena that aregenerated at the crack interface between steel and concrete and between the crack surfaces must be taken intoaccount through detailed and reliable laws formulated as a function of the crack kinematics.

In technical the literature several post-cracking R/C constitutive models have been proposed, which can be

divided into two different basic formulations: rotating and fixed crack models. Models based on smeared androtating cracks [4,7,8] allow for a gradual re-orientation of cracks along principal stress or principal straindirections and they describe the kinematics of crack through one parameter represented by the crack opening.Models based on smeared-fixed crack approach assume crack direction to remain fixed in the direction of firstcracking. These models, in order to satisfy the equilibrium conditions, have to add in their formulation theshear stresses that develop on crack surfaces and the shear slip [2,9]. Other models [1015], more complexbut with a more realistic approach, have been recently formulated on the assumption that cracks were discreteand fixed and the global behavior of the R/C element was due both to the contribution of uncracked R/Cbetween two adjacent cracks and of local stress and strain at crack interface, as well as of bond performancebetween concrete and reinforcement.

In this paper a macroscopic model [16], named 2D-PARC (two-dimensional physical approach for rein-forced concrete), which represents an advanced version of a previous model [14], based on realistic semi-empir-

ical constitutive laws for concrete, for reinforcing steel and for their interaction at the crack interface, andwhich is able to simulate the evolution of the crack pattern for plane stress R/C members, is proposed. Byassuming the behavior of uncracked concrete and reinforcement between contiguous cracks as that of twostructural elements working in parallel, while the behavior of uncracked R/C between cracks and the crackas that of two structural elements working in series, secant stiffness matrix is obtained in direct mode. Throughprogressive, nonlinear analysis up to failure, the model takes into account the parameters influencing primarycracking, that is the stress field and the orientation and spacing of reinforcing steel bars, and the parametersthat govern subsequent secondary cracking, that is, in addition to previous ones, bond between bars and con-crete, dowel action, aggregate bridging, aggregate interlock, degradation of concrete between cracks.

In order to verify the reliability and capability of the proposed approach, a comparison with observationsof a well-documented experimental tests [4,5,12] is carried out.

2. Modeling of reinforced concrete

2.1. Basic hypotheses

The proposed model describes the 2D mechanical behavior of a reinforced concrete membrane element sub-jected to general plane stresses (Fig. 1a). The theoretical formulation refers to a membrane element, with unit

Fig. 1. (a) Reinforced concrete membrane element: geometry and notation; (b) local co-ordinate system of the general ith steel bar layer

and (c) principal stress directions in the concrete.

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sides and thickness t, reinforced by ordinary steel bars arranged in n layers, each of them denoted by its axis xiforming an angle hi with respect to the global x-axis, by the cross-section area Asi, the diameter /i, the spacingsi, and smeared through the geometric steel ratio qi = Asi/(si t) (Fig. 1a and b).

By respecting the R/C physical conditions, the proposed model is able to simulate three different phases:uncracked, unidirectional cracking and bidirectional or multi-directional cracking.

2.2. Uncracked stage

With reference to the global xy co-ordinate system (Fig. 1a), in the uncracked phase perfect bond isassumed between steel and concrete; hence, concrete strain {ec} and steel strain {es} are equal to the totalstrain vector {e}:

fecg fesg feg: 1

Equilibrium conditions impose that the total stress {r} is the sum of the stresses in the concrete {rc} andthose in the steel {rs}. By equilibrium and compatibility conditions, as well as constitutive laws, it results:

frg frcg frsg Dcfecg Dsfesg Dc Dsfeg Dfeg; 2

where [D] = [Dc] + [Ds] is the global uncracked R/C stiffness matrix, [Dc] is the concrete stiffness matrix,Ds

Pni1Dsi; Dsi being the stiffness matrix of the ith steel bar layer and n the total number of reinforcing

layers.

2.2.1. Concrete stiffness matrix

The adopted constitutive law for uncracked concrete is based on [1719] and is defined in 12 local co-ordinate system, whose axes are directed along the principal maximum and minimum stress directions, respec-tively (Fig. 1c). Uncracked concrete is modeled as an orthotropic, nonlinear elastic material, being 12 theorthotropic axes, with stiffness matrix having the following expression:

D1;2c

1

1 m2

Ec1 m ffiffiffiffiffiffiffiffiffiffiffiffiffiEc1Ec2p 0mffiffiffiffiffiffiffiffiffiffiffiffiffiEc1Ec2

pEc2 0

0 0 1 m2Gc12

264 375; 3Ec1 and Ec2 being the concrete secant elastic moduli in the two orthotropic directions, and Gc12 the concreteshear modulus, assumed equal to:

Gc12 Ec1 Ec2 2m

ffiffiffiffiffiffiffiffiffiffiffiffiffiEc1Ec2

p41 m2

:

Ec1 and Ec2 are evaluated by describing the actual biaxial state of stress with two equivalent uniaxial states ofstress. The uniaxial curves are reported in Fig. 2a, where the peak stress rcmax and its strain ec are computed infunction of the ratio a = r1/r2 according to a biaxial strength envelope (Fig. 2b), [20].

In the global xy co-ordinate system the concrete stiffness matrix results:

Dc TuT

D1;2c Tu; 4

[Tu] being the transformation matrix, function of the u angle between the x-axis and the 1-axis (Fig. 1c).

2.2.2. Steel stiffness matrix

In the local xiyi co-ordinate system of each ith steel bar layer (Fig. 1b), the stiffness matrix Dxiyisi is eval-

uated by taking into account the axial and shear stiffnesses and strengths of steel bars, whose mechanicalbehavior is described by an elastic-hardening constitutive law (Fig. 3). Steel stiffness matrix takes the form:

Dxiyisi qsi

Esi 0

0 Gsi" #; 5

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where: Gsi Esi

21m. In xy co-ordinate system, the stiffness matrix is evaluated by adding the contribution of

each ith layer of reinforcing steel bars:

Ds Xni1

Dsi Xni1

TsiT

Dxiyisi Tsi; 6

[Tsi] being the transformation matrix, function of the hi angle.

2.3. Unidirectional cracking

Cracks are assumed to arise in concrete when the principal tensile stress (Fig. 1c) exceeds the tensilestrength. The crack is assumed to form at right angle with respect to principal tensile stress direction (thatis the current 1-axis described by the u angle, Fig. 1c). A local n1t1 co-ordinate system of the crack, wheren1 and t1 are perpendicular and parallel to crack direction, respectively, is introduced; this co-ordinate systemis described by the w1 angle (coincident with the current u angle) between n1 and x-axes (Fig. 4a and b).

In the crack local n1t1 co-ordinate system the mechanical quantities, associated with primary cracks, aredefined. Crack pattern is assumed as immediately fully developed with a crack spacing am1 which remains con-

stant during the loading process (Fig. 4a).

fc

= -0.06

= -0.15

=1.0

=0.52

=0.0

fct

c0c

maxc

0c

c

cs

c

0c

c

cs

c

c2

E

E1

E

E

+

=

c

c0

c

c0

0c

maxccsE

=

c1

c2

=

cmax

( ) ( ) ( )( ) ( )[ ] 12lim 1-k21h1k42-hk2-hk-

++=

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2tension-tension

c1

c2

=

c1=c2

tension-compression

compression- compression

c1=limc2

( )

lim

c2,max c2

c1,max c2,max

if 0

1 kf

1

+ =

+ =

lim

cc1,max

c2,max c1,max

c ct

if 1 1 0

f

h

f f ,

h 0.8 , k 3.65

=

+ = =

= =

compression failure

tension

failure

c1

c2

compression-tension

c2

c1

a b

Fig. 2. Assumed (a) equivalent uniaxial curves for tension and compression and (b) biaxial strength envelope.

c

s

sy

fsy

fsy

sy

su

su s

1

Es

1Esp

1sE

1

Es

1

Esp 1

sE

Fig. 3. Assumed elastic-hardening constitutive law for ordinary steel.


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After cracking, the total strain is constituted by two contributions, the first one, {ec}, related to theuncracked R/C, having a solid condition even if damaged, between two adjacent cracks and the secondone, {ecr1}, related to the kinematics developed in the crack. Therefore, it results:

feg fecg fecr1g; 7

where {ecr1} is the crack strain, which is defined in the crack local co-ordinate system n1t1 asfe

n1t1cr1 g

w1am1

v1am1

;w1 being the crack opening along the n1-axis and v1 being the crack slip along t1-axis

(Fig. 4b). The local crack strain vector is then transferred to the global xy co-ordinate system through thetransformation matrix Tw1 : fecr1g Tw1

1fen1t1cr1 g.

In cracked concrete, crack opening and slip activate several resistant mechanisms which provide strengthand stiffness. Some of these contributions are due to concrete, in particular to the aggregates acting upon

the crack surfaces (aggregate bridging and interlock), others are due to steel bars which cross the crack (ten-sion stiffening and dowel action). Hence, the crack stiffness matrix [Dcr1] is formed by adding the stiffnessmatrix due to concrete resistant contributions, [Dc,cr1], to the stiffness matrix due to reinforcement resistantcontributions, [Ds,cr1], as follows:

Dcr1 Dc;cr1 Ds;cr1: 8

With all the stiffening contributions of the crack formulated as a function of w1 and v1 (Fig. 4b), the equilib-rium in the crack, with reference to global xy co-ordinate system, yields to:

frg frcr1g frc;cr1g frs;cr1g Dc;cr1 Ds;cr1fecr1g Dcr1fecr1g; 9

where {rcr1} is the stress in the crack, {rc,cr1} and {rs,cr1} are the stresses in the crack balanced by resistantcontributions due to concrete (as aggregate bridging and interlock) and due to steel bars crossing the crack

(as tension stiffening and dowel action), respectively.The equilibrium condition in the uncracked R/C between two adjacent cracks can be written as:

frg frcg frsg Dcfecg Dsfesg; 10

where [Dc] and [Ds] are the same stiffness matrices defined for uncracked R/C, even if the terms of [ Dc] areadequately softened by a damage coefficient. As the average strain of steel between two contiguous cracks{es} has values little lower than the average strain {e} of element, here {es} = {e} is assumed.

From Eqs. (9) and (10), the crack strain and the strain of concrete between two adjacent cracks are obtainedas follows:

fecr1g Dcr11

frg

fecg Dc1

frg Dsfeg;

11

Fig. 4. (a) RC membrane element with unidirectional cracking: geometry and notation; (b) kinematical parameters of crack and (c)principal stress direction in the concrete between two adjacent cracks.


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which, substituted into the compatibility Eq. (7), yield to:

feg Dc1

frg Dsfeg Dcr11

frg Dc1

Dcr11

frg Dc1

Dsfeg;

and then:

frg Dc1 Dcr1

11I Dc

1Dsfeg Dfeg; 12

where:D Dc

1 Dcr1

1

1I Dc

1Ds 13

is the global R/C stiffness matrix and [I] is the identity matrix.

2.3.1. Stiffness matrix of the concrete between two adjacent cracks

The behavior of concrete between two adjacent cracks is similar to that assumed in uncracked stage, but adegradation due to the damaged material near the crack and due to the irregularity of crack spacing is takeninto account. This is implemented through a damage coefficient f [4,5,7], here assumed as f = (1 + 200 w/am1)

1, applied to the strength and to the stiffness of concrete.

2.3.2. Stiffness matrix of the steel reinforcement between two adjacent cracks

After cracking, in the R/C between two adjacent cracks steel bars retain axial stiffness but their shear stiff-ness rapidly softens owing to the low bond and slip between steel bars and surrounding concrete.

2.3.3. Concrete contributions to crack stiffness matrix: aggregate bridging and interlock

With reference to primary crack, the aggregate bridging is modeled by the following equation (Fig. 5a, [19]):

rb1 rct max

1 w1=w01p cb1

w1

am1 cb1e1; 14

where cb1 is the bridging coefficient, w01 is the crack opening corresponding to rb1 = 0.5rct max and p is a coef-ficient defining the shape of the curve. The parameters w01 and p are chosen according to the CEB-ModelCode 90 [21] bilinear law by imposing the same area under the curves, namely the fracture energy, in the rangefrom 0 to wc1, which is the crack opening corresponding to zero stress.

Aggregate interlock activates shear and normal stresses due to the slip between crack surfaces. This contri-bution is formulated by:ra1 c01c1

sa1 ca1c1:15

The ca1 and c01 coefficients are defined as functions of the crack opening and slip [22], (Fig. 5b):

ca1 s 1

ffiffiffiffiffiffiffiffiffiffi2w1Dmax

s a3 a4

v1w1

31 a4

v1w1

4 amw1 and c01 a1a2w2q1 1 v1

w1

2 qca1; 16

where a1a2 0:62; a3 2:45s

; a4 2:44 1 4s

; s 0:27fc and q = 0.25, Dmax being the maximum aggregate

size.

Fig. 5. (a) Aggregate bridging and (b) aggregate interlock actions.


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Finally, the concrete stiffness matrix in the crack co-ordinate system is obtained:

Dn1t1c;cr1

cb1 c01

0 ca1

!: 17

2.3.4. Steel contributions to crack stiffness matrix: tension stiffening and dowel actionIn the crack, the forces in the steel bars due to the axial stiffness and to the dowel action are modeled in the

local steel co-ordinate system (Fig. 6a) and then smeared so as to obtain the static equivalent stresses (Fig. 6b).Axial di1 and transversal gi1 components, with respect to the bar axis (local xiyi co-ordinate system), of thecrack displacement vector s1 are evaluated. Therefore, the matrix of stiffening contributions for the ith steelbar crossing the first crack is:

Dxiyisi;cr1 qsi

Ecr1si gi1 0

0 di1

" #; 18

where Ecr1si is the secant elastic modulus in correspondence of steel axial strain ecr1si at the crack (evaluated by

tension stiffening model proposed in the following), and gi1

ecr1si

di1 lsi is the stiffening coefficient, lsi

am1

coshiw1 isthe length of the ith bar between two adjacent cracks.The dowel action contribution is modeled according to [23], obtaining the following:

di1 13:66f0:38

c /0:25i

lsi

di1 0:2g0:64

i1

: 19

The stiffening contribution due to steelconcrete bond between to contiguous cracks is accompanied by a non-uniform distribution of the strain in the reinforcement. Through the equilibrium equations imposed for section(Fig. 7a), for concrete (Fig. 7b), for steel bar (Fig. 7c), and the compatibility equation:

ds

dx es ec; 20

where s is the slip between steel and surrounding concrete along bar axis, es and ec are the normal strain of steeland concrete along bar axis, respectively, the solving equation is obtained as follows:

d2sx

dx2

4

/Es1

Es

Ecq

ssx: 21

The bond-slip law of the Model Code 90 [21] (Fig. 7d) is adopted. The problem is numerically solved byusing the Finite Difference Method with the boundary conditions s(0) = 0 and s(ls/2) = d/2 (Fig. 7e). Then,the stress in the concrete are computed by integrating the concrete equilibrium equation, once the stresses

crs i

crs

crs i

crs

t1n1

w1

i1

crs

i1

crs

v1

xiyi

crs

b

crsiN

a w1

v1Sdi

l 2

amsiN

i1

i1

xiyit1n1

siN

Fig. 6. (a) Tension stiffening and dowel action of bar and (b) their smeared equivalent effect on crack surfaces.


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in the concrete at crack are known. Finally, after computing concrete strains, from Eq. (20) the steel strain inthe bar is obtained by imposing the symmetry conditions in x = 0 and x = ls/2 and by imposing that the meanvalue of steel strain computed between two adjacent cracks from the tension stiffening formulation is equal tothe steel strain evaluated in the global procedure. When the tension stiffening distribution satisfies the previousconditions, the steel strain along the bar is obtained (Fig. 7e), where escri is the steel strain of the ith reinforce-ment layer at the crack.

Afterwards, all the matrices Dxiyisi;cr1 are transferred to the crack n1t1 co-ordinate system and summed up,

obtaining the matrix Dn1t1s;cr1 . Then, the latter is added to D

n1t1c;cr1 obtaining D

n1t1cr1 that, finally, is transferred

to xy global co-ordinate system yielding [Dcr1]:

Dcr1 Tw1T

Dn1t1cr1 Tw1: 22

Fig. 7. Interaction between steel bar and concrete: (a) equilibrium condition of reinforced concrete element under tension; (b) of concretealone; (c) of steel bar alone; (d) adopted bond relationship (Model Code 90) and (e) shear bond stress and non-uniform distribution of steelbar tensile stress and strain.


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2.4. Bidirectional and multi-directional cracking

When in the concrete between contiguous cracks (Fig. 4c), considered in a solid condition even if damaged,the maximum principal tensile stress exceeds the tensile strength of concrete, secondary cracks form ( Fig. 8aand b). The secondary crack orientation is assumed to form a right angle with respect to the current principaltensile stress direction (current 1-axis described by the u angle, Fig. 4c). A local n2t2 co-ordinate system of thecrack, where n2 and t2 are perpendicular and parallel to secondary crack direction, respectively, is introduced;

this co-ordinate system is described by thew

2 angle (coincident with the currentu

angle) betweenn

2 andx-axes (Fig. 8c and d). The procedure by which stiffness matrix is evaluated is similar to that already seenfor the singly cracked R/C. The strain is decomposed into three contributions, the first one related to the con-crete between two adjacent cracks and the others related to the kinematics of the two cracks (Fig. 8c and d):

feg fecg fecr1g fecr2g: 23

Similarly to primary cracking stage (Eq. (13)), the global R/C stiffness matrix is:

D Dc1

Dcr11

Dcr21

1I Dc

1Ds

; 24

where [Dcr1] and [Dcr2] are the stiffness matrixes containing the resistant contributions of the first and second

crack system, respectively.In the general case of multi-directional cracking stage of R/C, being Nc the total number of occurred

cracks, the stiffness matrix is given by:

D Dc1

XNck1

Dcrk1

1I Dc

1Ds: 25

3. Comparisons with experimental observations

The capability of the proposed model to describe the evolution of crack pattern in R/C membrane elements

subjected to a general plane stress state is highlighted through the analysis of two panels (named PV10 and

Fig. 8. Reinforced concrete membrane element with bidirectional cracking: (a) geometry and notation and (bd) kinematical parametersof the primary and secondary cracks.


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PB21) tested in comprehensive and well-documented research projects [4,6,5,12]. In Table 1 the mechanicalcharacteristics of examined panels are reported.

Panel PV10, tested by Vecchio and Collins [4], was 890 mm square 70 mm thick, and it was reinforced bybars at right angle, aligned with the panel sides, having different steel ratio, Table 1, and subjected to mono-tonic pure shear, Fig. 9a.

Fig. 10 shows some images of the tested panel at various stages of loading: Fig. 10a shows the crack patternprior to yielding of transverse reinforcement (load stage 5) when there are only primary cracks oriented withan angle little more than 45 with respect to y-direction. Fig. 10b shows the crack pattern corresponding toultimate load capacity (load stage 7) with the appearance of secondary cracks oriented with an angle of about53 with respect to y-direction.

Fig. 11 shows some comparisons between experimental results and those predicted by the proposed model.The trend of the applied shear stress is reported as a function of the shear strain (Fig. 11a), of the principalstrain angle (Fig. 11b), of the longitudinal strain (Fig. 11c) and finally of the transversal strain (Fig. 11d).

Table 1Mechanical characteristics of examined panels

Panel Loading ratio sxy:rx:ry qsx fsx (MPa) qsy fsy (MPa) fc (MPa) fct (MPa) ec1

PV10 1:0.0:0.0 0.0179 276 0.0100 276 14.5 0.0027PB21 1:3.1:0.0 0.0220 402 21.8 2.4 0.0018

70 mm

x

y890 mm

yx

PV10

890mm

70 mm

x

y

x

890mm

PB21

yx

xy xy

yx

xy xy

x

890 mma b

Fig. 9. Geometrical characteristics of examined (a) PV10 [4] and (b) PB21 [5] panels.

Fig. 10. Crack pattern of panel PV10 at various stages of loading: (a) prior to yielding of transverse reinforcement (load stage 5) and (b) at

failure (load stage 7).


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Panel PB21 (Bhide and Collins, [5]) was 890 mm square 70 mm thick, containing only longitudinal rein-forcement, subjected to combined shear and uniaxial tension with loading ratio 1:3.1, Table 1.

The panel showed a considerable load carrying capacity beyond the cracking load. The initial cracks formedclose to the direction of predicted principal stresses, at about 71 to the reinforcement (Fig. 12a). As loadincreased, some cracks formed at about 50 and then others cracks formed at about 30 (Fig. 12b); the lattercracks were characterized by a rapid widening that caused the failure of the panel. The predicted response, com-

pared with experimental observations, is shown in Fig. 13, in terms of shear stress against shear strain

Shear strain (10-3)

0 2.5 5.0 7.5 10.0 12.5 150

1.5

3.5

4.5

0.5

4.0

2.5

Shearstressxy

(MPa)

1.0

2.0

3.0

2D-Parc

Experimental

42.5 47.545 52.5 55 57.5

Direction of principal strain (deg)40 50 60

0

1.5

3.5

4.5

0.5

4.0

2.5

Shearstres

sxy

(MPa)

1.0

2.0

3.0

2D-Parc

Experimental

Observed cracking

Longitudinal strain x (10-3)0-0.2 0.2 0.4 0.6 0.8 1.0

0

1.5

3.5

4.5

0.5

4.0

2.5

Shearstress

xy

(MPa)

1.0

2.0

3.0

2D-Parc

Experimental

1.2

Transversal strain y (10-3)0.0 2.0 4.0 6.0 8.0-2.0

0

1.5

3.5

4.5

0.5

4.0

2.5

Shearstress

xy

(MPa)

1.0

2.0

3.0

2D-Parc

Experimental

a b

c d

Fig. 11. Comparisons between observed and predicted responses of panel PV10: plots between applied shear stress versus (a) shear strain;(b) concrete principal strain angle; (c) longitudinal strain and (d) transverse strain.

Fig. 12. (a and b) Crack pattern of element PB21 subjected to combined shear and tension at different loading stages.


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(Fig. 13a), principal stress direction (Fig. 13b), longitudinal strain (Fig. 13c), and transversal strain (Fig. 13d).On the whole, the proposed model fits well the test results.

4. Conclusions

A consistent constitutive model able to describe the progressive behavior up to failure of reinforced con-crete plane stress elements under monotonic loading has been proposed. The model has been formulated interms of secant stiffness matrix for the implementation into finite element codes and it is suitable for analyzingthe nonlinear response of reinforced concrete structures and for evaluating service and ultimate loading con-ditions. The new characteristics of the extended model here presented, in comparison to those of the originalone [14], include: the use in the cracked phase of the same 2D constitutive law adopted for uncracked concrete,a more detailed law for tension stiffening, where a realistic nonlinear bond-slip law is adopted, a general for-mulation of the stiffness matrix for the cracked phase which allows a simple generalization for multi-direc-tional cracking phase. As a result, through this formulation it has been possible to obtain the completestrain and stress fields of the concrete and of the steel bars at the crack interface and between two adjacentcracks, as well as the crack pattern, the slip and opening of the crack for any load increment.

The comparisons between experimental observations and numerical predictions have shown a good agree-ment in terms of applied load versus shear strain, longitudinal and transversal strains, principal stresses andstrains in the concrete. In particular, the examined cases show that the proposed model is able to predict there-orientation of secondary cracks, thanks to the detailed description of interaction forces at the crack surface,which cause a change of the stress and strain fields of concrete between adjacent cracks with respect to those ofprimary cracks. This fact is well highlighted particularly in the analysis of PB21 specimen (which is subjected

to combined shear and uniaxial tension), because reinforcement is arranged only along one direction and shear

Shear strain (10-3)

0 2 4 6 8 10 120

0.50

1.00

1.50

0.25

1.25

0.75

Shearstress

xy

(MPa)

10 20 30 40 50 60 70 80 90

0 0.5 1.0 1.5

Longitudinal strain x (10-3) Transversal strain y (10-3)

0

1.0 2.0 3.0 4.0 5.00

0

Direction of principal stress (deg)

2D-ParcExperimental

0

0.50

1.00

1.50

0.25

1.25

0.75

Shearstress

xy

(MPa)

0.50

1.00

0.75

1.50

0.25

1.25

Shearstressxy(M

Pa)

0

0.50

1.00

1.50

0.25

1.25

0.75

Shearstressxy

(MPa)

2D-ParcExperimental

Observed cracking

2D-ParcExperimental

2D-Parc

Experimental

a b

c d

Fig. 13. Comparisons between observed and predicted response of panel PB21: plots between applied shear stress versus (a) shear strain;(b) concrete principal stress angle; (c) longitudinal strain and (d) transversal strain.


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stresses in the crack activate strong aggregate interlock and dowel action, producing a high slip of cracksurfaces.

The general approach here presented allows for some extensions which are currently under study. Thesemight include the case of non-monotonic and cyclic loadings (by introducing cyclic laws for describing theconstitutive relationships of materials and for the interactions between them), and the case of 3D modeling

of reinforced concrete.

Appendix A. Notation

Asi cross-sectional area of the ith steel reinforcement layer[D], [Dc], [Ds] reinforced concrete, concrete and steel stiffness matrices, respectively, evaluated in xy co-ordi-

nate system[Dcrk],[Dc,crk],[Ds,crk] stiffness matrices of all the resistant contributions, of the resistant contributions due to

concrete, of the resistant contributions due to the steel of the kth crack, respectively[Dsi] stiffness matrix of the ith steel layer evaluated in xy co-ordinate system

Dxiyisicrk matrix of stiffening contribution for ith steel bar crossing the kth crack in the local co-ordinate system

of the barEc1;Ec2;Gc12 concrete secant longitudinal and shear elastic moduli in the two directions of orthotropyEsi;Gsi steel secant longitudinal and shear elastic moduliEcrksi secant elastic modulus of the ith steel bar at the kth crackfc uniaxial compressive strengthfct uniaxial tensile strengthgik tension stiffening coefficient[I] identity matrixlsi length of ith bar between two adjacent cracksn total number of reinforcement layerssi spacing of bars of the ith layert thickness of membrane element

[Tsi], [Tu], [Twk] transformation matrices function of the hi angle between local xiyi co-ordinate system ofsteel bar ith layer and concrete principal stress co-ordinate system (Fig. 14b), function of the u anglebetween concrete principal stress system and xy co-ordinate system (Fig. 14a), and function of the wkangle between crack local nktk and global xy co-ordinate system (Fig. 14c and d), respectively

wk,vk opening and slip of the kth crack{e},{ec},{es} total strain, concrete strain and steel strain, respectively, evaluated in the global xy co-ordinate

systemfecrkg; fe

nktkcrk g crack strain evaluated in the global and in the local crack co-ordinate systems, respectively

esicrk steel axial strain at the kth crackf damage coefficientqi geometric ratio of the ith steel reinforcement layer

Fig. 14. Local co-ordinate systems: (a) principal stress directions for concrete; (b) direction ofith layer of steel bars and (c and d) normal

and parallel directions of the primary and secondary cracks, respectively.


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{r},{rc},{rs} applied stress state, stress state in uncracked concrete or in concrete between two adjacentcracks, stress state in the steel embedded in uncracked concrete or in concrete between two adjacentcracks, respectively

{rcrk},{rc,crk},{rs,crk} stresses in the crack interface, stresses in the crack interface balanced by resistant con-tributions due to concrete and due to steel bars crossing the crack, respectively

rbk,rak,sak normal stress due to aggregate bridging effect, normal and shear stresses due to aggregate inter-lock/i diameter of the ith steel reinforcement layer

References

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Engng 1985;12(3):62644.[4] Vecchio F, Collins MP. The modified compression-field theory for reinforced concrete elements subjected to shear. ACI J

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[10] Kaufmann W, Marti P. Structural concrete: cracked membrane model. J Struct Engng 1998;24(12):146775.[11] Vecchio FJ. Disturbed stress field model for reinforced concrete: formulation. ASCE J Struct Engng 2000;126(9):10707.[12] Vecchio FJ, Lai D, Shim W, Ng J. Disturbed stress field model for reinforced concrete: validation. ASCE J Struct Engng

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International Conference on Crack Paths CP. Parma 2006. Proc on CD.[17] Darwin D, Pecknold DA. Non-linear biaxial stressstrain law for concrete. J Engng Mech Division: Proc ASCE

1977;103(EM2):22941.[18] Elwi AA, Murray DW. A 3D hypoelastic concrete constitutive relationship. J Engng Mech Div: Proc ASCE 1979;105(EM4):62341.[19] Belletti B, Bernardi P, Cerioni R, Iori I. On a fiber reinforced concrete constitutive model for the NLFE analysis. Studies and

Researches, Milan University of Technology 2003;24:2350.[20] Kupfer H, Hilsdorf HK, Rusch H. Behavior of concrete under biaxial stresses. Proc ACI J 1969;66(8):65666.[21] CEB Bulletin dinformation No. 213/214. C.E.B.-F.I.P. Model Code 1990, 1993.[22] Gambarova PG. Sulla trasmissione del taglio in elementi bidimensionali piani di c.a. fessurati. Proc Giornate AICAP, 2629 May,

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