Sadeghi&Nouban Stress-Strain Law Concrete Cyclic

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INTERNATIONAL JOURNAL OfACADEMIC RESEARCH Vol. 2. No. 4. July 2010

A NEW STRESS-STRAIN LAW FOR CONFINEDCONCRETE UNDER CYCLIC LOADING

Dr. Kabir Sadeghi1and Fatemeh Nouban

2

1Engineering Faculty,

2Construction M. Department,

Girne American University, Girne (TRNC, CYPRUS)E-mails: [email protected], [email protected]

ABSTRACT

A new stress-strain formulation for confined concrete under cyclic loading is proposed. This law can beused in numerical simulations to describe the stress-strain relationship for fibers or layers within the sections ofreinforced concrete elements.

In order to eliminate the problem of scale effect, a finite element model was generated and the simulatedand experimental results of tests on the prototype reinforced concrete column models subjected to cyclic loadingwere compared. In this way, the proposed stress-strain law for confined concrete under cyclic loading wasaccordingly modified and then validated.

Key words: Non-linear, behavior, stress-strain law, confined concrete, monotonic, cyclic, loading.

1. INTRODUCTION

Several models have been proposed to determine the law governing the behavior of compressive concreteconfined within rectangular or spiral stirrup ties.

Among these models, the models of Kent and Park (1), Vallenas et al. (2), and Sheikh and Uzumeri (3)have been used for the calculation of the theoretical flexural resistance of columns. The details of these modelsare given in the references quoted above. It is to be noted that in all of these formulations, the enhancement ofductility due to confinement by stirrup ties are confirmed.

The significant lateral passive compression and strain on concrete due to the confinement generated bythe transverse reinforcements, indicates that the concrete is in the state of tri-axial compression. In this condition,the concrete element is very ductile.

The primary results published on this subject have shown that the resistance and longitudinal strain ofconcrete, confined by hydrostatic pressure activated by fluid, could be increased as defined by two simpleequations. These equations can be found in the literature (4, 5 and 6).

Researchers such as Mander et al. (7), Scott et al. (8), Sheikh and Uzumeri (9) and Vallenas et al. (2)have performed several tests on real full-scale structures.

A complex mathematical model was developed by Bazant and Bhat (10, 11). Apparently this model is thesole model that considers the stress-strain relation for the monotonic, cyclic and dynamic loading of confined orunconfined concretes under any state of multi-axial stresses. However this model has been developed by usingmainly the data based on the biaxial and tri-axial tests with the confinement provided by mechanical means.

Therefore, in this step of development, the passive confinement due to transverse reinforcement was nottaken into account. The primary researches on the behavior law of confined concrete were performed generally ata small scale on samples under the concentrated point loading by applying a quasi-static strain velocity.

The proposed model by Kent and Park (1) for the law to describe the behavior of concrete confined withinrectangular stirrup ties was based on the results of the experimental tests of Roy and Sozen (12) and also of

researchers who have taken into account the enhancement of ductility resistance.Scott et al. (8) and Park et al. (13) have also worked on the full-scale columns of buildings.By considering the augmentation of concretes resistance due to confinement and the effect of the strain

velocity, they proposed modification to the behavior law of Kent and Park (1).Vallenas et al (2), and Sheikh and Uzumeri (9) have proposed the monotonic stress-strain equation for

concrete confined by rectangular transverse reinforcements that take into account the above-mentionedconsiderations. Park et al. (14), Desayi et al. (15), Ahmad and Shah (16, 17), Meyer (18), Tassios (19) and othershave also proposed equations for the stress-strain law of concrete confined by spiral or circular transversereinforcements.

In the following a modification of one of the existing models for unconfined concrete is first presented. Aformulation for confined concrete under cyclic compression is then proposed by the authors.

2. EXPERIMENTAL DATA AND VERIFICATION METHODOLOGY2.1. Experimental data

The experimental test results on columns carried out at the University of Nantes by Garcia Gonzalez (20)are used for the global validation of the proposed model. Over 20 tests have been performed on prototype scaled


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columns under bi-axial alternating cyclic and axial loading. The horizontal loads have been applied throughdifferent horizontal directions on the top of the columns.

The experimental test results on the confined and unconfined concrete samples performed by Sadeghi(21) are also used for local validation.

Since unconfined concrete under cyclic loading and also confined concrete under monotonic loading arespecial cases for confined concrete under cyclic loading, the models of Al Sulayfani (22) (for unconfined concreteunder cyclic loading case) and Tassios (19) (for confined concrete under monotonic loading case) are modified

and used for validation of particular cases of the proposed model.The simulated results obtained by using the proposed model are also compared with the results reported

in literature.

2.2. Verification methodology

In order to eliminate the problem of scale effect, a finite element model entitled Biaxial Bending ColumnSimulation (BBCS) was developed by Sadeghi (23, 24 and 25) and the simulated and experimental results on theprototype column models were compared. In this way the proposed non-linear stress-strain law for confinedconcrete under cyclic loading was modified and validated.

2.3. Finite element model BBCS

In the BBCS program, a behavior law for confined concrete is considered and the behavior laws of

columns are specified. After an iterative process the simulation and experimental results are compared and theconfined concrete laws are adjusted accordingly. The selected laws, presented here are simple but sufficientlyaccurate to be used in a cyclic simulation.

The BBCS software deals with the simulation of a reinforced concrete column under biaxial bendingmoment and axial loading, for rectangular or non-rectangular sections, considering the non-linear behavior ofmaterials. The simulation method used in BBCS is convenient for cyclic loadings, such as earthquake excitationsor impacts against the reinforced concrete structures.

The basic equilibrium is justified over a critical hypothetical cross-section assuming the Navier law with anaverage curvature. The method used qualifies as a Strain Plane Control Process that requires the resolution of aquasi-static simultaneous equations system using a triple iteration process over the strains. It is based on thecyclic non-linear stress-strain relationships for concrete fibers and reinforcements. For this procedure, the strainsin the extreme compressive and tensile points of the section are considered as two main variables (for non-rectangular sections these points may outside the actual cross-sections). For the entire element, deflection isevaluated using an elasto-plastic analytic formulation.

The program takes into account the confining effect of the transverse reinforcement and simulates the loss

of the concrete cover. It allows the determination of the failure, the local internal behavior of critical section (i.e.strains, stresses, neutral axis,) and the global external behavior of the column (top deflection).

The main simulated results are confirmed with the full-scale experimental results obtained by otherresearchers (20, 32).

2.4. Examples of evaluation and modifying the parameters

Utilization of a formulation similar that of Al Sulayfani (22) for the simulation of a reinforced concreteelements behavior gives more closely spaced hystersis loops (moment-curvature loops) than are obtained by theexperimental tests on the columns of Garcia Gonzalez (20). To further investigate the problem, The models ofPark (26), Buyukozturk (27), Meyer (18), Darwin (28), Bazant and Kim (29) and Karsan and Jirsa have beenstudied (30).

Comparative studies on the parameters relating to the point of unloading (slope at the point of unloadingand the coordinates of residual stress) have been performed. By performing numerical simulation on the columns

using the BBCS program and comparing these with the experimental test of Garcia Gonzalez (20), it wasobserved that, the values of unloading tangent modulus EU0 and residual strain r (see Fig. 3) which are used forgenerating the stress-strain (-) curve play a very important role in the moment-curvature response of areinforced concrete section in the unloading phase. As shown in Fig. 3, the variation of E U is less important,because its effect is situated in the zone of lesser stresses.

The residual strain r has a non-negligible role in the failure phenomenon in the case of cyclic loading. Ifthis strain is considered greater than the actual case, the failure appears sooner in the response curve.Conversely, if a value of rsmaller than the actual is adopted, the number of cycles at failure and the values of thedisplacement and the curvature will reduce.

In Table 1, the variations of the values of 0r in function of 0A obtained by using different

researchers models are presented (see also Figs. 2 and 3).


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Table 1. Values of 0r obtained by using different models

0A 0.3 0.5 0.7 1.0 1.2 1.5 1.7 2.0

Al Sulayfani 0.019 0042 0.081 0.182 0.289 0.528 0.751 1.199

Meyer 0.024 0.068 0.132 0.340 0.442 0.615 0.744 0.960

Karsan et al. 0.052 .101 0.162 0.275 0.365 0.521 0.640 0.840Buyukozturk et al. 0.079 0.165 0.277 0.496 0.675 0.995 1.240 1.660

Based on the values given in this table, it can be observed that the smaller values of 0r are obtained

by using Al Sulayfanis model (for 0A


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The relation between the coordinates of the peaks of confined and unconfined concretes (19, 23),respectively are given as follows:

).50.2000.1( wccc ff for / fc < 0.05 or .w < 0.1 (2)

).25.1125.1( wccc ff for / fc 0.05 or .w 0.1 (3)

)(2

00f

f

c

ccc (4)

With: )(

max

).

(f

f

Sb

A

c

ytw

t

tk (5)

sn . (6)

3

81n (7)

)2

1(0

2

bSa ts for rectangular and circular sections (8)

)2

1(0b

Sa

ts for circular sections with spiral transverse reinforcements (9)

Where, is the stress, is the strain, fc and fcc are maximum compressive stresses of unconfined andconfined concretes, respectively, 0 and c0 are the strains at maximum unconfined and confined concretestresses, respectively, At is the cross sectional area of a transverse reinforcement, fyt is the yield stress oftransverse reinforcement, bmax is the larger dimension of section, St is the longitudinal spacing betweentransverse reinforcements, is the confinement efficiency factor = (confined area)/(total area), n is thetransverse reinforcements form factor, s is the transverse reinforcements spacing factor, and b0 is the distance

between external longitudinal reinforcements in the column section. The factors k and used in equations 5 and 7for some forms of transverse reinforcements (23, 19) are given in Table 2.

Table 2. Values of k and for some forms of transverse reinforcements (23, 19)

* K = 4 in case of circular transverse reinforcement.

85.0c 45.0c 0c

fcc

fcc

85.0

fcc45.0

L

Y

P

Fig. 2. Proposed stress-strain curve for confined concrete under monotonic loading(envelope curve for cyclic loading case)


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By inserting the relevant values of stress and strain in equation 1 at points L, P and Y and also recognizingthat the slope of the curve is equal to zero at the peak of curve (see Fig. 2), the four unknown coefficients A L, BL,CL and DL are determined.

These four conditions (coordinates of the points L, P and Y and also the slope at P) are found as follows:

Point L at coordinates (c0.45, 0.45 fcc) of the confined concrete stress-strain curve:

With: Ef Cccc 45.045.0 /)( 45.0 (10)

And secant modulus: fE ccc49.0

45.0 4861 (11) Point P at maximum stress of confined concrete stress-strain curve at coordinates (c0, fcc):

fccc246.0

0 00085.0 ( fcc in MPa) (12)

fc cc 0 (13)The supplementary condition of zero slope (tangent) at point P gives:

0

d

d(14)

Point Y at coordinates (c0.85, 0.85 fcc) on the confined concrete stress-strain curve:

wc 1.085.085.0 (15)fc cc85.085.0 (16)

where:

085.0 )9.1( 008.0 fcc ( fcc in MPa) (17)

3.2. Stress-strain law for confined concrete under cyclic loading3.2.1. Unloading curve

As shown in Figs. 3, 4 and 5, unloading may occur either from the envelope curve or from a phase ofreloading. In both cases, the equation 18 gives the stress-strain curve for unloading.

fDCBA ccUc

UUU

cc

)(

00

(0

23

)()

(18)

The four unknown factors AU, BU, CU and DU can be found by applying the coordinates and slopes of twoextreme points of unloading curve (i.e. the starting point of unloading and the extreme point on the unloadingcurve with zero stress).

a) Unloading from the envelope curveIn the case of unloading from the envelope curve (see Fig. 3), to find the unknown factors AU, BU, CU and

DU, the coordinates and slopes at points A(A, A) and B(r, 0) are used.

r

),( AAA

EU0

EU

),( 0rB

Fig. 3. Unloading from the envelope curve


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The tangent modulus at point A is given as follows:

)(3/)(100 rAAUE (19)The residual strain r at point B, depending on the unloading starting point coordinates (point A) can be

found from the equation 20 or 21.

)( 027.0

2

0

c

A

c

r

for 10

c

A

(20)

0

2

0

2.0

0

14.0 )(c

A

c

r

c

A for 10

c

A(21)

The modulus of the curve at point B can be found from the equation 22 or 23.

0

45.0 5.695.0

72.0

c

rc

U

E

E

for 6.1

0

c

r(22)

8.48

99.49

0

45.0

c

r

c

UE for 56.10

c

r (23)

b) Unloading from a phase of reloadingIn the case of unloading from a phase of reloading, depending on the coordinates of unloading point (C or

D) comparing with the coordinates of point A(A, A), different moduli are used as cited below and as shown inFigs. 4 and 5.

b1) Unloading from a point C where C < APoint C(C, C):The coordinates of point A have been determined in the previous step.The value of unloading tangent modulus (EUC) at point C is obtained by linear interpolation between the

modulus EU0 at point A and EU at point B which is given in equation 24.

)()( 0

0 rA

UUUUC

EEEE (24)

Point B(r, 0):The coordinates of point B have been determined in the previous step by applying the equation 20 or 21.The value of tangent modulus EU at point B has also been calculated by using the equation 22 or 23.

r

),( AAA

EUC

EU

),( CCC

)0,(rB

Fig. 4. Unloading from a point C on reloading curve when C< A

b2) Unloading from a point D where D > AFor the trajectory shown in Fig. 5, the two points D and E that the curve DE passes through them are

determined as follows:

Point D(D, D):The coordinates of point D have been determined in the previous step.The value of unloading tangent modulus (EUD) at point D is obtained by linear interpolating between the

modulus EU0 at point F and EU at point E.The point F is defined as the intersection of the line of slope 1.5E c0.45 passing from point D with the

envelope curve. By applying coordinates of point F instead of coordinates of point A, the equation 19 allows the

determination of the unloading modulus at point F. EUD can then be found by using equation 25:


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)(

)( 00

rF

UUUUD

EEEE

(25)

Point E(r, 0):The coordinates of point E are determined by applying F instead of A in the equation 20 or 21.The value of tangent modulus EU at point E is calculated by applying rof point E in the equation 22 or 23.

r

EUD

EU

FA

B E

D

Fig. 5. Unloading from a point D on reloading curve when D > A

3.2.2. Reloading curveEquation 26 is used for reloading curve:

fDCBA ccRc

RRR

cc

)(

00

(0

23

)()

(26)

The four unknown factors AR, BR, CR and DR can be found by applying the coordinates and slopes of thetwo extreme points of reloading curve (i.e. the starting point of reloading phase and the extreme point on theenvelope curve).

As shown in Figs. 6 and 7, reloading from zero stress and reloading from an unloading trajectory areconsidered in two different cases as follows:

a) Reloading from a zero stress status (residual strain)For the trajectory BG shown in Fig. 6, the two points B and G that the curve BG passes through them are

determined as follows:

Point B(r, 0):

The coordinates of point B have been determined in the previous steps.The value of reloading tangent modulus ER at point B can be obtained from equation 27:

0

45.0 78.295.0

1

c

rc

R

E

E

(27)

Point G(G, G):

With: AG 22.1 (28)The modulus ER0 is the tangent modulus on the envelope curve at point G.

r

),( AAA

),( GG

G

ER

ER0

)0,( rB

Fig. 6. Reloading from a point of zero stress

b) Reloading from an unloading trajectory

For the trajectory HI shown in Fig. 7, the coordinates and slopes of the two points H and I at the startingand finishing points of the curve HI are determined as follows:

Point H(H, H):


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The coordinates of point H have been determined in the previous step.The tangent modulus ERH at point H is considered to be equal to ER and can be calculated by using the

equation 27.

Point I(I, I):The strain at point I is calculated by using the equation 29.

rA

A

HrAI

KK

))1()(( (29)

where: K = 1.22

The strain of point I found from the equation 29 gives AI when AH and gives

AGI K (see equation 28 when rH (i.e. the point H is positioned on the point B).The stress I and modulus EI at point I can be found by using its strain and the envelope curve.

r

ER

ER0ERH

EI

B

A

H

I G

Fig. 7. Reloading from a point on the unloading trajectory

3.3. Application of the proposed formulationFig. 8 shows one example for the application of the proposed model to confined concrete. This Fig. shows

an example of concrete, confined within rectangular stirrup ties of diameter 8mm and a longitudinal spacing of6cm, under monotonic and cyclic loading.

Fig. 8.An example of confined concrete under monotonic and cyclic loading, fc= 42 MPa

3.4. The effect of variation in basic parametersFig. 9 shows an example obtained by application of the proposed behavior law. Variations in the spacing

St, of transverse reinforcements is evaluated.

Fig. 9. Influence of spacing Stof transverse reinforcements, fc = 42 MPa

It can be seen from Fig. 9 that:- the presence of the transverse reinforcements significantly increases the resistance capacity of the

reinforced concrete element;

0

5

10

15

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7

Strain (x 1000)

Stress(MPa

)

Cyclic

Monotonic

0

5

10

15

20

25

30

35

40

4550

0 1 2 3 4 5 6 7

Strain (x 1000)

Stress(MPa)

Spacing: 3 cm, Diameter: 6mmSpacing: 4 cm, Diameter: 6mmSpacing: 5 cm, Diameter: 6mmSpacing: 6 cm, Diameter: 6mmUnconfined Concrete


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- this influence is very important after achieving the maximum resistance capacity (after peak point).

3.5. Comparison of the proposed model and experimental tests resutsThe simulated results obtained by using the BBCS program have been compared with the results obtained

by Park et al. (32) on reinforced concrete elements under alternating mono-axial bending moment and theexperimental test of Garcia Gonzalez (20) on reinforced concrete elements under alternating bi-axial bendingmoment and axial force.

The simulated results using the BBCS program and the Park et al. results have been compared for arectangular section column with a cross-sectional area of 30.5cmx15.2cm under a vertical loading of 160 kN andan imposed alternating mono-axial bending moment. The section has 4 steel bars with a yield stress of fy = 345MPa and a total area of 11 cm

2, with a concrete cover of 5.1 cm. The characteristic resistance of concrete fc is

equal to 34 MPa.Comparison of the simulated and experimental moment-curvature curves, illustrated in Fig. 10, indicates a

close agreement between the proposed simulation and the experimental and simulation of Park et al. From thisFig. it can be observed that after the first half-cycle the newly proposed simulation is more precise than simulationof Park.

Fig. 10. Comparison of proposed model and experimental tests/simulation of Park (32)

4. CONCLUSIONS

The proposed stress-strain law for confined concrete under the compression cyclic loading is simple andapplicable to the simulation of reinforced concrete columns, beam-columns and beams under monotonic andcyclic loading.

The proposed law is validated by the results of experimental tests carried out on reinforced concretecolumns subjected to cyclic mono-axial and biaxial bending moment and axial load performed by differentresearchers.

Since the proposed law has a strong mathematical structure, it can readily be adapted and achieve ahigher degree of precision by modifying the relevant coefficients following the completion of more accurate andprecise tests. The proposed model can be used in conjunction with finite element analysis for the simulation of awide range of reinforced concrete elements under any loading conditions.

ACKNOWLEDGEMENTS

The financial and technical supports of University of Nantes/France, Water Resources ManagementOrganization/Ministry of Energy/Iran and The American University-Girne/Cyprus are appreciated.

REFERENCES

1. D.C. Kent and R. Park, Flexural members with confined concrete, Journal of the Structural Division,ASE, Vol. 97, no. ST7, Procs. Paper 8243: 1969-1990 (July 1971).

2. J. Vallenas, V.V. Bertero and E.P. Popov, Concrete confined by rectangular hoops subjected toaxial loads, report UCB/EE Rc-77/13, Earthquake Engineering Research Centre, University ofCalifornia: (1977).

3. Sh. A. Sheikh and S.M. Uzumeri, Properties of concrete confined by rectangular ties,AICAP-CEB

Symposium on structural concrete under seismic actions, Rome, (CEIB, Bulletin no.132): 53-60(1979).

-90

-60

-30

0

30

60

90

-0.06 -0.045 -0.03 -0.015 0 0.015 0.03 0.045 0.06

Curvature (1/m)

Moment(kNm

)

Num. Simul. (Park)

Propos. Num. Simul.

Experim. Test (Park)


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4. F.E. Richart, A. Brantzaeg and R.L. Broun, A study of the failure of concrete under combinedcompressive stress, Bulletin 185, University of Illinois, Engineering Experimental Station,Champaign, 111: (1928).

5. G.G. Balmer, Shearing strength of concrete under high tri-axial stress computation of Mohrsenvelope as a curve. SP-23, Structural Research Laboratory, U.S. Bureau of Reclamation: (1944).

6. F.E. Richart, A. Brantzaeg and R.L. Broun, The failure of plain and spirally reinforced concrete incompression, Bulletin 190, University of Illinois, Engineering Experimental Station, Champaign, 111:

(1929).7. J.B. Mander, M.J.N. Priestley and R. Park, Seismic design of bridges piers, Report no. 84-2, Dep. of

Civil Eng., Univ. of Canterbury, Christchurch: (1984).8. B.D. Scott, R. Park and M.J.N. Priestley, Stress strain behavior of concrete confined by overlapping

hoops at low and high strain rates,ACI Journal, 79(1): 13-27 (1982).9. Sh. A. Sheikh and S.M. Uzumeri, Strength and ductility of tied concrete columns, Journal of the

Structural Division, ASCE, Vol. 106, no. ST 5: 53-60 (1980).10. Z.P. Bazant and P.D. Bhat, End chronic theory of inelasticity and failure of concrete, Journal of

Engineering Mech. Division, ASCE, Vol.102, no. ST4: 701-722 (1976).11. Z.P. Bazant, P.D. Bhat, Prediction of hystersis of reinforced concrete members, Journal of the

Structural Division, ASCE, Vol.103, no. ST1: 153-167 (1977).12. H.E.H. Roy and M.A. Sozen, Ductility of concrete, Proc. Int. Symp. on the flexural mech. of

reinforced concrete, ASCE-ACI: 213-224 (1964).13. R Park, M.J.N. Priestley and W.D. Gill, Ductility of square-confined concrete columns, Journal of the

Structural Division, Proceedings of the ASCE, Vol. 108, no. ST4: 929-950 (1982).14. R. Park and P.D. Leslie, Curvature ductility of circular reinforced concrete confined columnsconfined by the ACI spiral, 6

thAustralian Conf. on the Mech. of Struct. & Mater., Vol. 1: 342-349

(1977).15. P. Desayi, K.T.S. Iyengar & T.S. Reddy, Equations for the stress-strain curve of concrete confined in

circular steel spiral, Materials and Structures Res. and testing, Paris, 11(65): 339-345 (1978).16. S.M. Ahmad and S.P. Shah, Stress-strain curves of concrete confined by spiral reinforcement,ACI

J., 79(6): 484-490 (1982).17. S.M. Ahmad and S.P. Shah, Behavior of hoop confined concrete under high strain rates, ACI J.,

82(5): 634-647 (1985).18. I.F. Meyer, Ein werkstoffgerechter schdigungsmodell und stababschnittselement fr stahlbeton

unter zyklicher nichtlinearer beanspruchung, SFB 151, Mitteilung no. 88-4: (1988).19. CEB, Confined concrete, CEB-FIP Model code 1990, Supplementary Documents for the first

predraft, Bulletin dinformation no. 189 (1988).20. J. Garcia Gonzalez, Contribution ltude des poteaux en bton arm soumis un cisaillement

dvi altern, Thesis of Doctorate (Ph.D.), Universit de Nantes: (1990).21. K. Sadeghi, Numerical simulation and experimental test of compression confined and unconfined

concrete, Technical report submitted to Water Resources Management Organization, Ministry ofEnergy, Concrete Laboratory of Power and Water University of Technology, Tehran: (2002).

22. B. Al Sulayfani, Contribution ltude du comportement des ossatures en bton arm soussollicitations cycliques par analyse non-linaire globale, Thesis of Doctorate (Ph.D.), Universit deNantes: (1986).

23. K. Sadeghi, Simulation numrique du comportement de poteaux en bton arm sous cisaillementdvi altern, Thesis of Doctorate (Ph.D.), Universit de Nantes: (1995).

24. K. Sadeghi, Proposition of a simulation procedure for the non-linear response of R/C columns orpiles under oriented lateral loading, International Journal of Engineering, vol. 5, no. 2a: 1-10 (1994).

25. K. Sadeghi, Proposition of a simulation procedure for the non-linear response of R/C columns undercyclic biaxial bending moment and longitudinal loading, Proceedings of The First InternationalConference on Concrete &Development, vol. 2, Tehran: 233-239 (2001).

26. R. Park and D.C. Kent, Sampson RA, Reinforced concrete members with cyclic loading, J. of theStructural Division, Procs. of ASCE, 9011 ST7: 1341-1360 (1972).

27. O. Buyukozturk and S.S. Shareef, Constitutive modeling of concrete in finite element analysis, J. ofComputers & Structures, Vol. 21, no. 3: 581-610 (1985).

28. D. Darwin and D.A. Pecknold, Nonlinear biaxial law for concrete, J. of Eng. Mech. Div. Vol. 105: 623(1979).

29. Z.P. Bazant and S.S. Kim, Plastic-fracturing theory for concrete, J. of Engineering MechanicsDivision, Proceedings of ASCE, Vol. 105, no. EM3: 407-429 (1979).

30. I.D. Karsan, J.O. Jirsa, Behavior of concrete under compressive loading, J. of the StructuralDivision, Proc. of ASCE, Vol. 95, 6935 ST12: 2543-2563(1969).

31. B.P. Sinha, K.H. Gerstle and L.G. Tulin, Stress-strain relation for concrete under cyclic loading,ACIJ., 61(2): 195-211 (1964).

32. R. Park, D.C. Kent and R.A. Sampson, Reinforced concrete members with cyclic loading, Journal ofthe Structural Division, proceeding of ASCE, ST7: 1341-1359 (1972).

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