HysteresisModels OTANI

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Chapter 11. Member Hysteresis Models

11.1 Introduction

An inelastic earthquake response analysis of structures requires realistic hysteresis models,

which can represent resistance-deformation relationship of a structural member model.

The resistance-deformation relations are different for constitutive materials of a section, for asection, for a member, for a story and for an entire structure. The resistance-deformation relation ofa structural analysis unit observed in a laboratory test must be idealized into aresistance-deformation hysteresis model. Different levels of resistance-deformation models must beused for structural elements considered in an analysis; e.g., a constitutive model of materials in afinite element method analysis, a hysteresis model for a rotational spring in a one-componentmember model, a story shear-drift hysteresis model for a mass-spring model.

A hysteresis model is derived by extracting common features of resistance-deformation relationsobserved in laboratory tests of members of similar properties. The hysteresis model of a membermust be able to express resistance-deformation relations under any loading history, including load

reversals.

Resistance-deformation relationship under monotonically increasing loading is called the primarycurve, skeleton curve or backbone curve. The skeleton curve provides an envelope of the hysteresisresistance-deformation relationship if the behavior is governed by stable flexure. The skeleton curvefor reinforced concrete member is normally represented by a trilinear relation with stiffness changesat flexural cracking and tensile yielding of longitudinal reinforcement. The skeleton curve of amember must be defined on the basis of mechanical properties of constitutive materials andgeometry of the member. Some researchers suggest the use of a bilinear relation with a stiffnesschange at yielding, ignoring the initial uncracked stage, because a reinforced concrete membersubjected to light axial force can be easily cracked by shrinkage or accidental and gravity loading.

The state-of-the-art does not provide a reliable method to estimate the initial stiffness, yielddeformation and ultimate deformation. The stiffness degrades from the initial elastic stiffness withincreased inelastic deformation and the number of cycles under reversed loading. The elasticmodulus of concrete varies significantly with concrete strength and mix; initial cracks cause decay inthe stiffness. The estimate of yield deformation is more complicated by the interaction of bendingand shear deformation and additional deformation due to pullout of longitudinal reinforcement fromthe anchorage zone and due to bar slip of longitudinal reinforcement along the longitudinalreinforcement within the member. Empirical expressions are necessary for the estimate of yield andultimate deformation.

The coordinates of a response point on a deformation-resistance plane are given by (D, F), inwhich, D: deformation, F: resistance. The skeleton curve is represented by either "bilinear" or

"tri-linear" lines for a reinforced concrete member, with stiffness changes at "cracking (C)" and"yielding (Y)" points.

The following terms are defined to clarify the hysteresisdescription;

Loading: a case where the absolute value of resistance (ordeformation) increases on the skeleton curve;

Unloading: a case where the absolute value of resistance(or deformation) decreases after loading or reloading; and

Reloading; a case where the absolute value of resistance(or deformation) increases after unloading before theresponse point reaching the skeleton curve.

The hysteresis model is formulated on the basis of resistance-deformation relations observed inthe laboratory tests. The loading program for a test should include the followings;

(1) At least two cycles of load reversals at an amplitude to study the decay in resistance at the

Loading

Unloading

Reloading

D

F


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amplitude,(2) Small deflection amplitude

excursion must be placed after a largeamplitude excursion to study theslip-type behavior

A lateral load-deflection relation of areinforced concrete member wasobtained from the test of a slendercolumn (Otani and Cheung, 1981). Thebehavior was dominantly by flexurealthough flexural cracks started toincline due to the presence of highshear stresses before flexural yielding.The yielding of the longitudinalreinforcement was observed in cycle 3.

The general hysteretic characteristics

can be summarized as follows:(a) Stiffness changed due to the flexural cracking of concrete and the tensile yielding of the

longitudinal reinforcement (cycle 1);(b) When a deflection reversal was repeated at the same newly attained maximum deformation

amplitude, the loading stiffness in the second cycle was noticeably lower than that in the first cycle,although the resistance at the peak displacement was almost identical (cycles 3 and 4). Thisreduction in stiffness is attributable to the formation of new cracks during loading cycle 3, and also toa reduced stiffness of the longitudinal reinforcement in cycle 4 due to the Bauschinger effect.

(c) Average peak-to-peak stiffness of a complete cycle decreases with previous maximumdisplacement. Note that the peak-to-peak stiffness of cycle 5 is significantly smaller than that of cycle2, although the displacement amplitudes of the two cycles are comparable. The peak-to-peakstiffness of cycle 5 is closer to that of cycles 3 and 4;

(d) The hysteresis characteristics of reinforced concrete are dependent on the loading history,and

(e) The resistance at the peak deflection is almost the same for the two successive cycles in themember dominated by flexural behavior.

A hysteresis model of a reinforced concrete "flexural" member must be able to represent theabove characteristics. The skeleton curve is similar to an "envelope curve" of a force-deformationrelation under load reversals. The state of the art is not sufficient to determine the ultimate point, atthe deformation of which the resistance of a member starts to decay. The force-deformation relationafter the onset of strength decay is normally not modeled because the behavior is stronglydependent on a particular local deterioration of materials.

If the reinforced concrete is subjected to

high shear stress reversals, or if theslippage of the reinforcement from concretewithin the anchorage area occurs, theforce-deflection curve exhibits a pronounced"pinching". The pinching behavior is alsoobserved;

(a) in a "flexural" member when theamount of longitudinal reinforcement differssignificantly for the tension and compressionsides at the critical sections, typically in agirder with monolithically cast slabs,

(b) at a member end where additional

deformation may be caused by anchorageslip of longitudinal reinforcement within theadjacent member or connection, and

Hysteresis of slip type (Bertero and Popov, 1977)


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(c) in a member where bond splitting cracks develop along the longitudinal reinforcement.

Because such hysteresis relationship is highly dependent on loading history and structuralproperties of the member, a general hysteresis model is difficult to formulate; or the parameters ofhysteresis models cannot be analytically determined by the properties of the member. In the designof earthquake resistant structures, the pinching type behavior is generally thought to be undesirable

because small hysteresis energy can be dissipated by the behavior. Therefore, a proper design caremust be exercised to reduce such pinching behavior due to shear and bond deterioration.

Many hysteresis models have been developed in the past. Some hysteresis models are elaborate,and include many hysteresis rules; others are simple. The complicatedness of a hysteresis modelindicates a large memory to store the hysteresis rule program in a computer. It does not lead to alonger computation time because the complicatedness of a hysteresis model requires simply manybranches in a computer program, and only a few branches are referred to for a step of responsecomputation.

A class of hysteresis models, in which the unloading and reloading relation is defined by

enlarging the skeleton curve by a factor of two, are called "Masing type." Some examples of Masingtype models are shown below:

A hysteresis energy dissipation index (Eh) isused to express the amount of hysteresis energy

dissipation W Δ per cycle during displacementreversals of equal amplitudes in the positive andnegative directions;

mm

h

DF

W E

π 2

Δ=

in which Fm: resistance at peak displacement Dm.The value of the index was derived by equating

the area of hysteresis and the energy W Δ dissipated by an equivalent viscous damper of alinearly elastic system in one cycle under the"resonant" "steady-state" oscillation.

The steady state response amplitudem

D

under sinusoidal excitation with amplitude o p

and circular frequency , is given by

Hysteresis energy dissipation index


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)sin()(

)(4})(1{

1

222

φ ω

ω

ω

ω

ω

+=

+−

=

t Dt x

hk

p D

m

nn

o

m

The energy dissipated W Δ by viscous damper per cycle is

2

2

22

0 0

2

2

)(cos))((

m

m

T T

m

Dmk h

Dc

dt t Dcdt dt

dx

dt

dxcW

n n

ω π

ω π

φ ω ω

=

=

+==Δ ∫ ∫

where k cm ,, ,: mass, damping coefficient and stiffness of an SDF system, h : damping factor

(

k m

c

2

= ), nT : natural period of the system (

k

mπ 2= ), n : circular frequency of the system

(m

k = ).

At the resonant condition (n

= ), the energy dissipated per cycle can be expressed2

2 m Dk hW π =Δ

Therefore, the damping factor corresponding to the hysteresis energy dissipation W Δ is

m

m

mmm

D

F k

DF

W

Dk

W h

=

Δ=

Δ=

π π 22 2

The equivalent damping factor should not be confused with a damping factor of a viscouslydamped system because the equivalent damping factor is not relevant in random oscillation.

References:

Bertero, V. V., and E. P. Popov, "Seismic Behavior of Ductile Moment Resisting ReinforcedConcrete Frames," ACI SP-53, American Concrete Institute, Detroit, 1977, pp. 247-291.

Comite Euro-International du Beton: RC Frames under Earthquake Loading, State of the Art Report,Thomas Telford, 1996.Otani, S, "Hysteresis Models of Reinforced Concrete for Earthquake Response Analysis," Journal,

Faculty of Engineering, University of Tokyo, Vol. XXXVI, No. 2, 1981, pp. 125-156.Otani, S., and V. W.-T. Cheung, "Behavior of Reinforced Concrete Columns Under Bi-axial Lateral

Load Reversals - (II) Test Without Axial Load," Publication 81-02, Department of CivilEngineering, University of Toronto, 1981.

Saatcioglu, M., "Modeling Hysteretic Force-Deformation Relationships for Reinforced ConcreteElements," ACI-SP127, American Concrete Institute, Detroit, 1991, pp. 153-198.


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11.2 Bilinear Model

At the initial development stage of nonlinear dynamic analysis, the elastic-perfectly plastichysteretic model ("elasto-plastic model") was used by many investigators. The response pointmoves on the elastic stiffness line before the yield stress is reached. After yielding, the responsepoint moves on the perfectly plastic line until unloading takes place. Upon unloading, the response

point moves on the line parallel to the initial elastic line.

This model does not consider degradation of stiffness under cyclic loading. Energy dissipationduring a small excursion is not included.

A finite positive slope was assigned tothe stiffness after yielding to simulate thestrain hardening characteristics of thesteel and the reinforced concrete("bilinear model"). Unloading stiffnessafter yielding is equal to the initial elasticstiffness. The stiffness degradation with

inelastic deformation and energydissipation during small amplitudeoscillation are not considered in themodel.

Neither the elasto-plastic model northe bilinear model represents thebehavior of reinforced concrete and steelmembers. The steel member softensduring reloading after plastic deformationby the "Bauschinger effect." Theresponse of the elasto-plastic model is

compared with a test result of areinforced concrete column above.

When the degradation in stiffness was recognized in the behavior of the reinforced concrete, theloading and unloading stiffness K r was proposed to degrade with the previous maximumdisplacement (Nielsen and Imbeault, 1970) in a form:

α −= )(

y

m

yr

D

DK K

in which, α : unloading stiffness degradation parameters (0 < α


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constant until the response displacement amplitude exceeds the previous maximum displacement ineither direction. The model is called a "degrading" bilinear hysteresis model." If the value of a ischosen to be zero, the unloading stiffness does not degrade with yielding. A smaller value of a tendsto yield a larger residual displacement. The degrading bilinear model does not dissipate hystereticenergy until the yield is developed. For a reinforced concrete member, the value of α is normally

selected to be around 0.4.

The hysteretic energy dissipation index E h ofthe degrading bilinear model is given by

)1)(1(

)}1(){1(2α

α

βμ μβ β μ π

μβ β μ μ β

−+−

+−−−=

h E

in which β : ratio of the post-yielding stiffness

to the initial elastic stiffness; and : "ductility

factor" (ratio of the maximum displacement tothe initial yield displacement).

The equation is valid for a ductility factorgreater than 1.0. The hysteresis energy index ofa regular bilinear model (α = 0) reaches ashigh as 0.33 at a ductility factor of 4.0. However,such large amplitude oscillations do notcontinue during an earthquake; no hysteresisenergy is dissipated by the model during smallamplitude oscillations. The total energydissipation of the bilinear model over theduration of an earthquake is much smaller thanthat expected from the hysteretic energydissipation index.

Reference:

Nielsen, N. N., and F. A. Imbeault, "Validity ofVarious Hysteretic Systems," Proceedings,Third Japan National Conference onEarthquake Engineering, 1971, pp. 707-714.


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11.3 Ramberg-Osgood Model

A stress-strain relation of the metal was expressed using three parameters y D , yF and γ by

Ramberg and Osgood (1943), where y D : yield displacement, yF : yield resistance and γ : a

parameter of the model. Jennings (1963) introduced the fourth parameter η to the model. The

initial loading curve of the model under monotonically increasing deformation, as modified byJennings, is expressed by

)1(

1−

+=

γ

η y y y

F

F

F

F

D

D

in which, γ : exponent of the Ramberg-Osgood model; and η : parameter introduced by Jennings

(1963).

The initial tangent modulus is equal to (F y/ D y), and the initial loading curve passes a point (F y,(1+η ) D y) for any value of γ . The shape of the primary curve can be controlled by the exponent γ

from linearly elastic (γ = 1.0) to elasto-plastic (γ = infinity). For a larger value of γ , the behavior

becomes similar to that of the bilinear model.

Upon unloading from a peak response point ( Do, F o), the unloading, load reversal and reloadingbranches of the relationship is given by

)2

1(22

1−

−+

−=

− γ

η y

o

y

o

y

o

F

F F

F

F F

D

D D

until the response point reaches the peak point of one outer hysteresis loop.

The resistance F is not explicitly expressed by a given displacement D in this model. Theresistance F at a given displacement D must be computed numerically, for example, using theNewton-Rapson's iterative procedure.

The Ramberg-Osgood model is often used for stress-strain relation of the steel in the finiteelement analysis or in the lamina model, and for resistance-deformation relation of steel members ina frame analysis.

The hysteresis energy dissipation index of the Ramberg-Osgood model is expressed as

)1)(121(2

m

m

y

yh

DF

F

D E −+−= γ

η π

The model can dissipate some hysteresis energy even if the ductility factor is less than unity. The


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index is sensitive to the exponent γ

of the model, and the hysteresisenergy dissipation capacity increaseswith increasing value of the exponent.

References:

Jennings, P. C., "Response of SimpleYielding Structures to EarthquakeExcitation," Ph.D. Thesis,California Institute of Technology,Pasadena, 1963.

Ramberg, W., and W. R. Osgood,"Description of Stress-StrainCurves by Three Parameters,"National Advisory Committee on Aeronautics, Technical Note 902,

1943.


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11.4 Degrading Tri-linear Model

A model that simulates dominantly flexural stiffness characteristics of the reinforced concrete wasused extensively in Japan (Fukada, 1969). The primary curve is of tri-linear shape with stiffnesschanges at flexural cracking and yielding. Up to yielding, the model behaves in a manner the sameas the bilinear model. When the response exceeds a yield point, response point follows the

strain-hardening part of theprimary curve. Onceunloading takes place froma point on the primary curve,the unloading point isconsidered to be a new"yield point" in the direction.The model behaves in abilinear manner betweenthe positive and negative"yield points" with stiffnessdegraded proportional to

the ratio of the slopesconnecting "current yieldpoints" and "the initial yieldpoints."

The ratio of the first and second stiffness is kept constant even after yielding.

This model has the following properties:(a) the stiffness continuously degrades

with increasing maximum amplitude beyondyielding,

(b) the hysteretic energy dissipation is

large in the first load reversal cycle afteryielding, and becomes steady in the followingcycles, and

(c) the steady hysteretic energy dissipationis proportional to the displacement amplitude.

The hysteretic energy dissipation index ofthe degrading tri-linear model is expressed as

y

c

c

y

hF

F

K

K E )1(

2−=

π

in which K y: secant stiffness at yielding (=

y y DF / ), and cK : initial elastic stiffness (=cc DF / ). The index is independent of the

displacement amplitude, but dependent onthe stiffness and resistance ratios at crackingand yielding. Cracking point of this modelcontrols the fatness of a hysteresis loop.Therefore, it is important to choose thecracking point taking into account the degreeof a hysteresis loop.

Nomura (1976) used an arbitrary skeletoncurve; when the response point reached the previous maximum response point, it moves on the

skeleton curve. Upon unloading, the newly attained maximum response point was considered as theyield point in the direction, similar to the degrading tri-linear model.

Degrading tri-linear model


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References:

Fukada, Y., "Study on the Restoring ForceCharacteristics of Reinforced Concrete Buildings(in Japanese)," Proceedings, Kanto Branch

Symposium, Architectural Institute of Japan, No.40, 1969, pp. 121-124.

Nomura, S., "Restoring Characteristics and theirModeling," Data for Earthquake ResistantDesign for Buildings, No. 65, Magazine of Architectural Institute of Japan, June 1976.

Nomura model (1976)


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11.5 Clough Degrading Model

A hysteretic model with an elasto-plasticskeleton curve was proposed by Cloughand Johnston (1966) to represent thehysteretic behavior of a reinforced concrete

beam-column sub-assemblage.

During loading, the response pointfollows the elasto-plastic skeleton curve.The unloading stiffness after yielding waskept equal to the initial elastic stiffness. Theresponse point during reloading movestoward the previous maximum responsepoint in the direction of reloading, simulatingthe stiffness degradation. If yielding has nottaken place in the direction of reloading, theresponse point moves toward the yield point

in the reloading direction.

A minor deficiency of the Clough modelwas pointed out by Mahin and Bertero(1976). After unloading from point A,consider a situation in which reloading takesplace from point B. The original Cloughmodel assumed that the response pointshould move toward the previous maximumresponse point C. This is not realistic.Therefore, a minor modification was addedso that the response point should move

toward an immediately preceding unloadingpoint A during reloading. When theresponse point reaches the point A, theresponse point moves toward the previous maximum point C.

The model was made more versatile by incorporating the reduction in unloading stiffness K r witha maximum displacement in a form:

α −= )( y

m yr

D

DK K

in which, α : unloading stiffness

degradation parameter; yK : initial elastic

stiffness; and m D : previous maximumdisplacement. The different unloading

stiffness may be assigned taking m D to be

a maximum deformation in the directionunloading takes place.

If the value of a is chosen to be zero, theunloading stiffness of the model remainsequal to the initial elastic stiffness.

The response of the Clough model isshown to compare well with the response ofa reinforced concrete column tested in thestructures laboratory.

Clough Model

RC Column

Column Top Displacement, mm

C o l u m n R e s i s t a n c e ,

k N

-100 -50 0 50 100

100

200

0

-100

-200

D

F

B

C

Y

Y

K r =K y

K y

A

Clough Model

D

F

B

C

Y

Y

K r

Ky

Dm D y

F y A

Modified Clough Model


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Saiidi and Sozen (1979) and Riddell and Newmark (1979) used models similar to the modifiedClough model.

Wang and Shah (1987) introduced the strength and stiffness degradation effect of cumulativedamage. The strength and stiffness degrade in proportion to (1- Dws), where Dws is the Wang andShah damage index. The ordinates of the bilinear skeleton curve in monotonic loading is multiplied

by the current value of (1- Dws). Unloading and reloading stiffness is reduced by the same amount, asthey are defined on the basis of the location of the point of reversal and of the maximum previousdeformation in the direction of loading, on the degraded skeleton curve. The Wang and Shahdamage index is defined separately for each direction of loading as

1

1

n

ws n

e D

e

δ −=

−

where the damage prameter δ is expressed in terms of chord rotation,

i

i

u

c

θ

δ θ

=∑

The hysteretic energy dissipation indexof the modified Clough model is expressedas

})1(

1{1

μ

μ μβ β

π

α +−−=

h E

where β : ratio of post-yielding stiffness to

the initial elastic stiffness, and : ductility

factor.

The equation is valid for ductility factorgreater than unity. The Clough model cancontinuously dissipate hysteretic energyeven at a small amplitude oscillation afteryielding.

References:

Clough, R. W., and S. B. Johnston, "Effectof Stiffness Degradation onEarthquake Ductility Requirements,"Proceedings, Second Japan NationalConference on Earthquake

Engineering, 1966, pp. 227-232.Mahin, S. A., and V. V. Bertero, "Rate of

Loading Effect on Uncracked and Repaired Reinforced Concrete Members," EERC No. 73-6,Earthquake Engineering Research Center, University of California, Berkeley, 1972.

Riddell, R., and N. M. Newmark, "Statistical Analysis of the Response of Nonlinear Systemssubjected to Earthquakes," Structural Research Series No. 468, Civil Engineering Studies,University of Illinois at Urbana-Champaign, Illinois, 1979.

Saiidi, M., and M. A. Sozen, "Simple and Complex Models for Nonlinear Seismic Response ofReinforced Concrete Structures," Structural Research Series No. 465, Civil Engineering Studies,University of Illinois at Urbana-Champaign, Illinois, 1979.

Wang, M.-L., and S. P. Shah, “Reinforced Concrete Hysteresis Model based on the DamageConcept,” Earthquake Engineering and Structural Dynamics, John Wiley & Sons, Chichester,

Sussex, Vol. 15, 1987, pp. 993 -1003.


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11.6 Takeda Degrading Model

Based on the experimental observation on the behavior of a number of medium-size reinforcedconcrete members tested under lateral load reversals with light to medium amount of axial load, ahysteresis model was developed by Takeda, Sozen and Nielsen (1970). The model has been widelyused in the nonlinear earthquake response analysis of reinforced concrete structures.

Takeda Hysteresis Model:1. Condition: The cracking load, Pcr , has not been exceeded in one direction. The load is reversed

from a load P in the other direction. The load P is smaller than the yield load P y.Rule: Unloading follows a straight line from the position at load P to the point representing the

cracking load in the other direction.

2. Condition: A load P1 is reached in one direction on the primary curve such that P1 is larger thanPcr but smaller the yield load P y. The load is then reversed to -P2 such that P2P1.Rule: Unloading follows a straight line joining the point of return and the point representing

cracking in the other direction.

4. Condition: One or more loading cycles have occurred. The load is zero.Rule: To construct the loading curve, connect the point at zero load to the point reached in the

previous cycle, if that point lies on the primary curve or on a line aimed at a point on the primarycurve. If the previous loading cycle contains no such point, go to the preceding cycle and continuethe process until such a point is found. Then connect that point to the point at zero load.

Exception: If the yield point has not been exceeded and if the point at zero load is not locatedwithin the horizontal projection of the primary curve for that direction of loading, connect the point atzero load to the yield point to obtain the loading slope.

5. Condition: The yield load P y is exceeded in one direction.Rule: Unloading curve follows the slope given by the following equation:

4.0)( D

Dk k y

yr =

in which r k : slope of unloading curve, yk : slope of a line joining the yield point in one direction to

the cracking point in the other direction, D : maximum deflection attained in the direction of the

loading, and y D : deflection at yield.

6. Condition: The yield load is exceeded in one direction but the cracking load is not exceeded inthe opposite direction.

Rule: Unloading follows Rule 5. Loading in the other direction continues as an extension of theunloading line up to the cracking load. Then, the loading curve is aimed at the yield point.

7. Condition: One or more loading cycles have occurred.Rule: If the immediately preceding quarter-cycle remained on one side of the zero-load axis,

unload at the rate based on rule 2, 3 and 5 whichever governed in the previous loading history. If theimmediately preceding quarter-cycle crossed the zero-load axis, unload at 70% of the rate based onrule 2, 3, or 5, whichever governed in the previous loading history, but not at a slope flatter than theimmediately preceding loading slope.

Takeda model included (a) stiffness changes at flexural cracking and yielding, (b) hysteresis rulesfor inner hysteresis loops inside the outer loop, and (c) unloading stiffness degradation with

deformation. The response point moves toward a peak of the one outer hysteresis loop. Theunloading stiffness K r after yielding is given by


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α −

+

+=

y

m

yc

yc

r D

D

D D

F F K

in which, α : unloading stiffness degradation parameter; and m D : previous maximum displacement

beyond yielding in the direction concerned. The hysteresis rules are extensive and comprehensive.

The hysteretic energy dissipation index ofthe Takeda model is expressed as

})1(

1

1

1{1

μ

μβ β μ

π

α +−

+

+

−=

y

c

y

c

h

F

F

D

D

E

The expression is valid for a ductility factor

greater than unity.

It should be noted that the Takedahysteresis rule was originally developed tosimulate the behavior of reinforced concretemembers. If this model is used to simulate thebehavior of a story or a simplified structure,some rules need to be simplified.

For example, hysteresis rules prior toyielding may be simplified such that unloadingtakes place toward the origin of the relation

(Muto Model). This model is often used in astory-based (mass-spring) earthquakeresponse analysis.


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Bilinear Takeda Model: The primary curve of the Takeda model can be made bilinear simplychoosing the cracking point to be the origin of the hysteretic plane. Such a model is called the"bilinear Takeda" model, similar to the Clough model except that the bilinear Takeda model hasmore hysteresis rules for inner hysteresis loops (Otani and Sozen, 1972); i.e., the response point

moves toward an unloading point on the immediately outer hysteresis loop.

The behavior before yielding issometimes made simple by letting theresponse point moves toward theorigin during unloading, and towardthe maximum response point in theopposite side upon reloading. TheTakeda hysteresis rules are appliedafter the yielding.

This model is similar to Clough

Degrading Model, but is morecomplicated having rules for innerhysteresis loops.

Additional modifications of the Takeda model with bilinear backbone curve may be found inliterature (Powell, 1975, Riddle and Newmark, 1979, Saiidi and Sozen, 1979, Saiidi, 1982). Riddleand Newmark (1979) used a bilinear skeleton curve and unloading stiffness equal to the initial elasticstiffness; loading occurs either on the strain hardening branch or towards the furthest point attainedin the previous cycle. Saiidi and Sozen (1979) claimed to simplify the Takeda model using a bilinearskeleton curve; the model, however, is identical to the modified Clough model with reducedunloading stiffness with maximum deformation, and reloading to the immediate prior unloading pointif reloading occurs during unloading and then to the unloading point on the skeleton curve.

References:

Takeda, T., M. A. Sozen and N. N. Nielsen, "Reinforced Concrete Response to SimulatedEarthquakes," Journal, Structural Division, ASCE, Vol. 96, No. ST12, 1970, pp. 2557-2573.

Otani, S., and M. A. Sozen, "Behavior of Multistory Reinforced Concrete Frames DuringEarthquakes," Structural Research Series No. 392, Civil Engineering Studies, University ofIllinois, Urbana, 1972.

Powell, G. H., “Supplement to Computer Program DRAIN-2D,” Supplement to Report, DRAIN-2DUser’s Guide, University of California, Berkeley, August 1975.

Riddle, R., and N. M. Newmark, “Statistical Analysis of the Response of Nonlinear Systemssubjected to Earthquakes,” Structural Research Series No. 468, Civil Engineering Studies,

University of Illinois, Urbana, 1979.Saiidi, M., “Hysteresis Models for Reinforced Concrete,” Journal, Structural Division, ASCE, Vol. 108,

No. ST5, May 1982, pp. 1077 - 1087.Saiidi, M., and M. A. Sozen, “Simple and Complex Models for Nonlinear Seismic Response of

Reinforced Concrete Structures,” Structural Research Series No. 465, Civil Engineering Studies,University of Illinois, Urbana, 1979.

D

F

Dm

D’m X0

(D0,F0)

X1

(D1,F1)

X3

(D2,F2)

(D3,F3)


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11.7 Pivot Model

Major features of the force-deflection hysteresis results of large-scale reinforced concretemembers are;

(1) Unloading stiffness decreases as displacementductility increases,

(2) Following a nonlinear excursion in one direction,upon load reversal, the force-deflection path crosses theidealized initial stiffness line prior to reaching theidealized yield force, and

(3) The effect of pre-cracked stiffness may beignored.The use of the pivot point in defining degraded unloadingstiffness was first proposed by Kunnath et al. (1990).

Four quadrants are defined by the horizontal axisand the elastic loading lines (positive and negative).Primary Pivot points P1 through P4 on the elastic loading

lines control the amount of softening in each quadrant;points P1 and P4 on positive elastic stiffness line andpoints P2 and P3 on the negative elastic stiffness line. The resistance of primary pivot points P3 and

P4 is 2 2 yF α and the resistance of primary

pivot points P1 and P2 is 1 1 yF α .

Pinching Pivot points PP2 and PP4 fix thedegree of pinching following load reversal ineach quadrant. The resistance of pinching

pivot point PP4 is*

1 1 yF β and the resistance

of pinching pivot point PP2 is*

2 2 yF β .

(1) The response follows the strengthenvelope so long as no displacementreversal occurs. The strength envelope isdefined by the initial elastic stiffness, yieldresistance Fy, top point T, degradation pointD and failure point F, and can be different inpositive and negative directions.

(2) Once the yield deformation has beenexceeded in either direction, a subsequentstrength envelope is developed requiring the

introduction of upper bound points S1 and S2 which move along the strength envelope and defined by the previous maximum displacements. Theinitial points of S1 and S2 are yield point Y1 and Y2 in each direction. The strength envelope isdefined by lines joining PP4 and S1 and points PP2 and S2.

(3) The modified strength envelope (acting as the upper bound for future cyclic loading) is defined bylines joining the pinching pivot point PP4 (PP2) to maximum response point S1 (S2) until the responsepoint reaches the strength envelope.

(4) The pinching pivot points PP4 and PP2 are initially fixed, but they move toward theforce-deflection origin with the strength degradation. The resistance at a pinching pivot point is given

by*

i yiF β where

Q1

Q2

Q3

Q4 4 2 2( )

y

P F α

2 1 1( ) yP F α

Y2(Dy2,Fy2)

Y1(Dy1,Fy1)

* *

4 1 1 1 1( , ) y yPP D F β β

* *

2 2 2 2 2( , ) y yPP D F β β

Q1

Q2

Q4

Q3

D

F

Y1

Y2

P1

P4 P3

P2

dy dt1 dd1 df1

Fy1

Fy2

dy2

2 2 yF α

1 1 yF α

dt2dd2df2

PP4

PP2

Ft1

Ft2

D1

T1

F1

T2

D2

F2

*

1 1 yF β

*

2 2 yF β


17/57

17

)(

)(

*

*

tiiMAX

ti

iMAX

ii

tiiMAX ii

d d F

F

d d

>=

≤=

β β

β β

where i β defines the degree of pinching

for a ductile flexural response prior tostrength degradation. tiiMAX d d , : maximum

displacement and strength degradationdisplacement (displacement at the highestresistance) in the i-th direction of loading(i=1 or 2).

Hysteresis Rules:(1) Loading and unloading in Quadrant Qn (n=1 or 3) is directed away from or toward point Pn,respectively.Modification (Otani): Loading in Quadrant Qn (n=1 or 3) is directed toward maximum response point

Si, followed by the strength envelope. Unloading in Qn (n=1 or 3) is directed toward point Pn.(2) Loading in Quadrant Qn (n=2 or 4) is directed toward point PPn, then to maximum response pointSi, followed by the strength envelope.(3) Unloading in Quadrant Qn (n=2 or 4) is directed away from point Pn.

Q1

Q2

Q4

Q3

D

F

Y1

Y2

P1

P4 P3

P2

PP4

PP2

S1

S2

Q1

Q2

Q4

Q3

D

F

Y1

Y2

dt1 df1

dt2df2

PP4

PP2

D1

T1

F1

T2

D2F2

S1

S2

D1MAX

D2MAX

F2MAX

F1MAX

Ft2

Ft1


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18

Modification for Softened Initial Stiffness:The unlading stiffness of the maximum

displacement excursion in Quadrant Q1 is guidedtoward point P1. A new pivot point P1* is defined

on this unloading line at force (1 η + ) times largerthan the force at point P1. A line extending frompoint P1* through origin defines the new softenedelastic loading line K*. Point PP4* is defined by theintersection of the modified strength envelope(line between points PP4 and S1) and the newsoftened elastic loading line K*.

Reference:

Dowell, R. K., F. Seible and E. L. Wilson,“Pivot Hysteresis Model for ReinforcedConcrete Members,” ACI StructuralJournal, Title No. 95-S55, Vol. 95, No. 5,September-October 1998, pp. 607 -617.

Kunnath, S. K., A. M. Reinhorn, and Y. J.Park, “Analytical Modeling of InelasticSeismic Response of RC Structures,”Journal, Structural Engineering Division, ASCE, Vol. 116, No. 4, April 1990, pp.996 - 1017,”

Q2 Q3

Q4 Q1

S1

D

PP4

PP4*

1 1 yF α

1 1(1 ) yF η α +

P1 P1*

F P4 P4*

K*


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19

Pivot Hysteresis Model (Version 2)

Reference:Dowell, R. K., F. Seible and E. L. Wilson, "Pivot Hysteresis Model for Reinforced Concrete

Members," ACI Structural Journal, Title No. 95-S55, Vol. 95, No. 5, September-October 1998, pp.607 - 617.

Modification:(1) Initial stiffness is the same for positive (IS=2) and negative (IS=1) directions.(2) Loading in Quadrant Qn (n=1 or 3) is directed toward maximum response point Si, followed by thestrength envelope.(3) Unloading in Quadrant Qn (n=1 or 3) is directed toward point Pn.(4) Loading in Quadrant Qn (n=2 or 4) is directed toward point PPn, then to maximum response pointSi, followed by the strength envelope.(5) Unloading in Quadrant Qn (n=2 or 4) is directed away from point Pn.

Hysteresis Rules:Rule 1: Loading on strength envelope in positive or negative direction.Rule 2: Unloading from maximum response point X(IS) on strength envelope toward pivot point

PV3(3-IS) on the other side or loading toward maximum point X(IS) on the same side.Rule 3: Loading toward pivot point PV2(IS) on the same side after Rule 2 crossing displacement axis

at displacement D2.Rule 4: Unloading from Rule 3 at point (D3,F3) away from pivot point PV3(IS) on the same side until

the response point crosses displacement axis at displacement D4 or loading towardunloading point (D3,F3) followed by Rule 3.

Rule 5: Loading toward maximum response point X(IS) on the same side after Rule 3 passing pivotpoint PV2(IS).

Rule 6: Unloading from Rule 5 at point (D5,F5) toward pivot point PV3(IS) on the other side until theresponse point crosses displacement axis at displacement D2, or loading toward unloadingpoint (D5,F5) followed by Rule 5.

Rule 7: Loading toward maximum response point X(IS) after Rule 4 crossing displacement axis atdisplacement D4.

Rule 8: Unloading from Rule 7 at point (D5,F5) toward pivot point PV3(IS) on the other side untilresponse point crosses displacement axis at displacement at D2, or loading towardunloading point (D5,F5) followed by Rule 7.

Y4

F

Y1

Y1

PV3(IS)

PV3(IS)

PV2(IS)

PV2(IS)

X(IS)

X(IS)

Rule 1

Rule 2

D2Rule 3

Rule 4

Rule 5

Rule 2

Rule 3

Rule 4 D4

Rule 7

Rule 7

Rule 6

Y2

Y3

Y2

Y3

Y4

Rule 1

Rule 8

Negative Direction IS=1

Positive Direction IS=2

D2

(D3,F3)

D4

(D5,F5)

(D5,F5)


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Rule 9: Loading initial elastic stiffness after loading on Rule 3 passing pivot point PV2(IS), orunloading on initial stiffness before crossing displacement axis at origin.

Rule 10: Loading on the initial elastic stiffness before pivot point PV2(IS) is reached followed by Rule5 or unloading on the initial elastic stiffness before until the response point crosses thedisplacement axis at the origin followed by Rule 9.

Rule 1: Loading on strength envelope in Quadrant 1 or 3.

Rule 2: Unloading from maximum response point X(IS) on strength envelope towardpivot point PV3(3-IS) on the other side or loading toward maximum point X(IS)on the same side.

F

Y1

Y3

PV3

PV3

X(DX,FX)

X(DX,FX)

Rule 1

T

T

LV1

LV2

LV3

LV4

Y2

Y3

Y4

Y1

Y2

LV1

LV2LV3

LV4

For each Yi and LVi

(DYi,FYi) and SYi

Rule 2

Rule 2

D2D2Y4

Negative Direction

Positive Direction

Rule 1

Q2

Y4

F

Y1

Y1

PV3(IS’)

PV3(IS’)

PV2(IS)

PV2(IS)

X(DX,FX)

X(DX,FX)

Rule 1

Rule 2

D2Rule 3Rule 2

Rule 3D2

Negative Direction

Positive Direction

Rule 1

Rule 2


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21

Rule 3: Loading toward pivot point PV2(IS) on the same side after Rule 2 crossingdisplacement axis at displacement D2.

Rule 4: Unloading from Rule 3 at point (D3,F3) away from pivot point PV3(IS) on thesame side until the response point crosses displacement axis at displacementD4 or loading toward unloading point (D3,F3) followed by Rule 3.

Y4

F

Y1

Y2

PV3(IS)

PV3(IS)

PV2(IS)

PV2(IS)

X(DX,FX)

X(DX,FX)

D2

Rule 3

Rule 4

Rule 3

Rule 4 D4

D4

D2

(D3,F3)

(D3,F3)

Negative Direction

Positive DirectionRule 5

Rule 5

Rule 3

Y4

F

Y1

Y2

PV3

PV3

PV2

PV2

X(DX,FX)

X(DX,FX)

Rule 3

Rule 4

Rule 3

Rule 4 D4

D4(D3,F3)

(D3,F3)Rule 7

Rule 7

Negative Direction

Positive Direction

Rule 4


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22

Rule 5: Loading toward maximum response point X(IS) on the same side after Rule 3passing pivot point PV2(IS).

Rule 6: Unloading from Rule 5 at point (D5,F5) toward pivot point PV3(IS) on the otherside until the response point crosses displacement axis at displacement D2, orloading toward unloading point (D5,F5) followed by Rule 5.

D

F

Y1

Y2

PV3(IS’)

PV3(IS’)

PV2

PV2

X(DX,FX)

X(DX,FX)

Rule 5

Rule 5

Rule 6

Rule 6

Rule 1

Rule 1(D5,F5)

(D5,F5)

D2D2

Negative Direction

Positive Direction

Rule 5

D

F

Y1

Y2

PV3(IS)

PV3(IS)

PV2

PV2

X(DX,FX)

X(DX,FX)

Rule 5

Rule 5

Rule 6

Rule 6

Rule 1

Rule 1(D5,F5)

(D5,F5)

D2D2

Negative Direction

Positive Direction

Rule 3

Rule 3

Rule 6


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Rule 7: Loading toward maximum response point X(IS) after Rule 4 crossingdisplacement axis at displacement D4.

Rule 8: Unloading from Rule 7 at point (D5,F5) toward pivot point PV3(IS) on the otherside until response point crosses displacement axis at displacement at D2, or

loading toward unloading point (D5,F5) followed by Rule 7.

D

F

Y1

Y1

PV3(IS)

PV3(IS)

PV2(IS)

PV2(IS)

X(IS)

X(IS)

Rule 3

Rule 3

D4

D4

Rule 7

Rule 7

(D5,F5)

D2

(D5,F5)

Rule 8

Rule 8

D2

Negative Direction

Positive Direction

Rule 8

D

F

Y1

Y1

PV3(IS)

PV3(IS)

PV2

PV2

X(IS)=(DX,FX)

X(IS)=(DX,FX)

D4

D4

Rule 7

Rule 7

(D5,F5)

D2

(D5,F5)

Rule 6

Rule 6

D2

Negative Direction

Positive Direction

Rule 1

Rule 1

Rule 7


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Rule 9: Loading initial elastic stiffness after loading on Rule 3 passing pivot pointPV2(IS), or unloading on initial stiffness before crossing displacement axis atorigin.

Rule 10: Loading on the initial elastic stiffness before pivot point PV2(IS) is reachedfollowed by Rule 5 or unloading on the initial elastic stiffness before until theresponse point crosses the displacement axis at the origin followed by Rule 9.

F

Y3

PV3

PV3

Y1(DY,FY)

X(DX,FX)

Rule 9

T

T

LV1

LV2

LV3

LV4

Y2

Y3 Y4

Y1Y2

LV1

LV2LV3

LV4

For each Yi and LVi

(DYi,FYi) and SYi

D2D2 Y4

Negative Direction

Positive Direction

Rule 3PV2

Rule 9

Rule 9

PV2Rule 10

D2

Rule 1

F

Y3

PV3

PV3

Y1(DY,FY)

X(DX,FX)

Rule 10

T

T

LV1

LV2

LV3

LV4

Y2

Y3 Y4

Y1Y2

LV1

LV2LV3

LV4

For each Yi and LVi

(DYi,FYi,SYi)

D2D2 Y4

Negative Direction

Positive Direction

PV2

Rule 9

PV2

Rule 10

Rule 5


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11.8 Stable Hysteresis Models with Pinching

The force-deformation relation of a reinforced concrete member is highly dependent on a loadinghistory, characterized by strength decay with load reversals and pinching behavior at a low stresslevel during reloading, when the behavior of the member is dominated by sliding along inclined shearcracks or slippage of longitudinal reinforcement.

A flexure-dominated reinforced concretegirder sometimes exhibits a pinchingcharacteristic when the amount oflongitudinal reinforcement (or bendingresistance) is significantly different at the topand bottom of section. This is attributable tothe fact that a wide crack in weak sidecannot close due to large residual strain intensile reinforcement after load reversal; thecompressive stress must be resisted by thelongitudinal reinforcement before concrete

faces make contact at cracks.

Many hysteresis models have beendeveloped on the basis of test results of aparticular set of specimens under a specificloading history. However, the parameters of most models may not be analytically defined by themember properties (material properties and member geometry).

Takeda-slip Model: Eto and Takeda (1973) modified the Takeda model to incorporate a slip-typebehavior at low stress level due to pull-out of longitudinal reinforcement from the anchorage zone.

The skeleton curve is tri-linear with stiffness changes at cracking and yielding where the crackingand yielding levels can be different in positive and negative directions. The performance of themodel is identical to the Takeda model before yielding.

Pinching takes place only when theyielding has occurred in the direction ofreloading. The reloading (pinching)stiffness K s is defined as

m ms

m o y

F DK

D D D

γ −

=−

where o D : displacement at the end of

unloading (resistance equal to zero),

m D and mF : maximum deformation and

resistance in the direction of reloading,

y D : yield deformation in the direction of

reloading, γ : slip stiffness degradation

index (slip stiffness degradation index γ

is suggested to be 0.5). The pinchingstiffness is revised only when themaximum response point is exceeded inthe direction of reloading.

When the response point crosses a line connecting the origin and the maximum response point inthe direction of reloading, the response point moved toward the previous maximum response pointand then on the skeleton curve. The unloading stiffness is defined in the same manner as the

D

F Y( Dm ,F m)

K d o D

C

C

Y

K s

K s’

(Dm’,Fm’)

'o D

Takeda-slip model

Hysteresis Relation of Beams withUnbalanced Amount of Reinforcement


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Takeda model.

The same pinching and unloading stiffness is used during reloading and unloading in an innerloop.

'

'

c y md

c y y

F F DK

D D D

α −+

=

+

where, 'cF and 'c D : resistance and deformation at cracking on the opposite side, yF and y D :

resistance and deformation at yielding on the unloading side, m D : maximum deformation on the

unloading side, α : unloading degradation index.

Kabeyasawa-Shiohara Model: Kabeyasawa et al. (1983) modified the Takeda-Eto slip model torepresent the behavior of a girder with the amount of longitudinal reinforcement significantly differentat the top and bottom;


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(1) the pinching occurs only in one direction where the yield resistance is higher than the otherdirection,

(2) the pinching occurs only after the initial yielding in the direction of reloading, and(3) the stiffness K s during slipping is a function of the maximum response point ( Dm, F m) and the

point of load reversal ( Do, F o=0.0) in the resistance-deformation plane.

The reloading (slip) stiffness K s, after unloading in the direction of the smaller yield resistance,was determined as

γ

om

m

om

ms

D D

D

D D

F K

−−=

where (mm

F D , ): deformation and resistance at the previous maximum response point,o

D :

displacement at the end of unloading on the zero-load axis, γ : slip stiffness degradation index. No

slip behavior will be generated for γ = 0; the degree of slip behavior increases with γ > 1.0. γ =

1.2 was suggested.

The slip stiffness is used until the response point crosses a line with slope K p through the

previous maximum response point ( Dm, F m); the stiffness is reduced from the slope connecting theorigin and the maximum response point by reloading stiffness index η ,

)(m

m p

D

F K η =

The values of unloading stiffness degradation index α of Takeda model, slipping stiffnessdegradation index γ , and reloading stiffness index η were chosen to be 0.4, 1.0 and 1.0,

respectively by Kabeyasawa et al. (1983).

Costa and Costa model: Costa and Costa (1987) proposed a

trilinear model for the force-displacement response of asingle-degree-of-freedom oscillator, including pinching andstrength degradation.

Unloading-reloading loops prior to yielding in either directionare bilinear, with slopes equal to those of the pre-cracking andpost-cracking branches in the virgin loading. After the initial

yielding, the reloading stiffness sK is reduced from the stiffness

toward the previous extreme point by factor ( / ) y m D D γ

; i.e.,

( ) ym

s

m o m

DF K

D D D

γ =−

where, mF and m D : resistance and deformation at the

previous maximum response point, and o D : deformation at

load reversal point. Once the response point crosses the lineconnecting the origin and the maximum response point, thenresponse point moves toward the maximum response point.

The unloading stiffness after yielding is reduced from the

elastic stiffness by factor ( / ) y r D D α

.

Post-yield strength and stiffness degradation with cycling is

modeled by directing the reloading branch, after modification forpinching, toward a point at a displacement equal to (1 ) m Dλ +

D

F

C

Y

K s

O

Dc D y

F c

F y

F’c

F’ y

D’c D’ y


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28

and at a moment (1 ) mF λ − , where mF is the resistance at the extreme point if the previous

excursion. After reaching this terminal point of the reloading branch, further loading takes placeparallel to the post-yielding stiffness of the virgin loading curve.

References:

Costa, A. C., and A. G. Costa, “Hysteretic Model of Force-Displacement Relationships for Seismic Analysis of Structures,” National Laboratory for Civil Engineering, Lisbon, 1987.

Eto, H, and T. Takeda, "Elasto Plastic Earthquake Response Analysis of Reinforced ConcreteFrame Structure (in Japanese)," Report, Annual Meeting, Architectural Institute of Japan, 1973,pp. 1261-1262.

Kabeyasawa, T., H. Shiohara, S. Otani and H. Aoyama, "Analysis of the Full-scale Seven-storyReinforced Concrete Test Structure," Journal of the Faculty of Engineering, the University of

Tokyo, (B), Vol. XXXVII, No. 2, 1983, pp. 431-478.

F’c

F’ y

D’c

D’ y

Dc

D y

F c

F y


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11.9 Shear-type Hysteresis Models

Reinforced concrete members exhibit progressive loss of strength under reversed cycles ofinelastic deformation due to lack of shear capacity of member or bond resistance along longitudinalreinforcement; the monotonic strength of such members cannot be attained.

The response of a reinforced concrete member, exhibiting early strength decay, is difficult tomodel because such behavior is sensitive to loading history. General features can be summarizedas the decay in resistance with cyclic loading and pinching response during reloading followed byhardening.

The undesirable features can be avoided or reduced by following design requirements anddetailing of reinforcement. Therefore, hysteresis models for shear-failing performance may not benecessary for the response analysis of new construction, but may be necessary for the seismicevaluation of existing construction.

Takayanagi-Schnobrich Model: Takayanagi and Schnobrich (1976) modified the Takeda model to

incorporate pinching and strength decay features caused by high shear acting in short couplingbeams connecting parallel structural walls. The skeleton curve is trilinear.

The reloading (loading in the opposite direction after unloading) is made smaller than the stiffnesstoward the previous maximum response point in the direction of reloading; the response point movestoward the previous maximum response point after the response deformation changes its sign.

The resistance at a target point for reloading in the hardening range is reduced from theresistance at the previous maximum response point; e.g., the resistance at the target point isselected on a strength decay guideline which descends from the yield point. After the responsereaches the target point, the response point moves along a line parallel to the post yielding line.

The pinching stiffness is based on the reinforcement resistance for bending. The rate of strengthdecay is assumed to proportionally increase with the rotation.

Roufaiel-Meyer Model: Roufaiel and Meyer (1987) used a hysteresis model that includes strengthdecay, stiffness degradation and pinching effect.

Pinching

Decay Guideline

M

Mc

Mc

My

My

θ Dm

Dm’

Y’

Y

Takayanagi-Schnobrich Model of

Pinching and Strength Decay


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The moment resistance of a bilinearmoment-curvature relation was assumed to decaywhen a given strain is reached at the extremecompression fiber. The curvature at thecommencement of strength decay is called thecritical curvature. The degradation in resistance was

assumed to be proportional to the amount by whichthe critical curvature was exceeded.

An auxiliary unloading branch AB is drawnparallel to the elastic branch of the bilinear skeletoncurve until it intersects a line OB through the origin Oparallel to the strain-hardening branch YA of theskeleton curve. The line connecting this latter point Bof intersection to the point of previous extremedeformation in the opposite direction defines the endC of the unloading branch on the horizontal axis. Ifyielding has not taken place in the direction of

loading, the yield point is used as the previousmaximum response point.

From that point on reloading is not always directed straight to the point of the previous extremepost-yield excursion in the direction of reloading, but it may include pinching, depending on the shearration, M/Vh. Pinching is accomplished by directing the reloading branch first towards a point on theelastic branch of the skeleton curve at an ordinate equal to that of the intersection of this branch withthe line of straight reloading to the previous extreme deformation point, times m


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31

3

2

( ) { ( ) } y

p f f

f y

m half cycle at m mφ φ

φ φ φ φ

⎛ ⎞−Δ = − ⎜ ⎟⎜ ⎟−⎝ ⎠

Accordingly, a branch of reloading in the direction where the previous maximum curvature is equal to

φ , moves toward a point at ( ( ) , p

m mφ φ − Δ ), rather than at ( ( ), pm φ φ ) as in the original Roufaiel

and Meyer model.

Banon-Biggs-Irvine Model: Banon, Biggs and MaxIrvine (1981) modified Takeda hysteresis model by (a)using a bilinear skeleton curve, (b) incorporatingpinching and stiffness degradation. The pinchinghysteresis was adopted to simulate the propagation ofinclined cracks due to high shear and slippage oflongitudinal reinforcement.

Hysteresis rules are summarized below;

(a) Moment-rotation relationship is elastic up to theyield point,

(b) Once the yield point is exceeded, loadingproceeds on the second slope of the bilinear envelope,

(c) Unloading is parallel to the elastic stiffness,(d) The stiffness during reloading immediately after

unloading is reduced to 50 % of the second slope of thebilinear envelope,

2

2s

K K =

(e) When the direction of loading changes during unloading and resistance (or deformation)starts to increase again, the reloading stiffness is parallel to the elastic stiffness before the response

point reaches a point where the last unloading started,(f) When the sign of deformation changes during reloading, the response point moves toward

previous maximum response point in the direction of reloading.

If the strength-degrading feature is introduced, the response point after the pinching does notmove toward the previous maximum point, but a point on the skeleton curve at deformation greaterthan the previous maximum deformation.

* mm D

Dα

=

and 0.8α = is suggested in the study.

The skeleton curve may be different in positive and negative directions.

Kato Shear Model: Kato et al. (1983) used ahysteresis model to represent the behavior of areinforced concrete member failing in shear, inwhich strength decay and stiffness reduction due toload reversals were incorporated. A trilinear skeletoncurve was used with stiffness changes at A and B.By choosing the skeleton curve without descendingstiffness, the stable flexural behavior may berepresented by this model. The trilinear skeletoncurve may include descending slopes. The followingexample shows a skeleton curve with twodescending slopes.

F

D

Y

Y’

Dm D’m

sK

1K

2K

Banon-Bi s-Irvine Model 1981

D

F

A

B

Skeleton Curves of Kato Model (1983)


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32

The response is linearly elastic before the response point reaches point A. The response pointfollows the skeleton curve if the slope of the skeleton curve is positive; if the slope of the skeletoncurve is negative, the response point increases its deformation without the change in resistance(plastic behavior).

If a response point crosses the descending branches during loading or reloading, the deformation

increases without change in resistance (perfectly plastic stiffness). Upon unloading from a maximumresponse point on the perfectly plastic branch, the response point moves on a line parallel to the

initial elastic stiffness eK until the response point crosses the descending skeleton curve; the point

is termed as the maximum response point ( Dmax, F max). Then the response point follows a line with

reduced stiffness uK ;

max( )u e y

DK K

D

α −=

where α : unloading stiffness degradation index, y D : yield deformation.

Upon reloading after crossing zero resistance line, the response point moves on a line with

reloading (slip) stiffness sK ;

maxmin

min

( )so y

DF K

D D D

β −=−

where ( minmin , F D ): previous maximum response point on the skeleton curve in the direction of

reloading, o D : deformation at the completion of unloading, y D : yield deformation in the opposite

direction.

This slip stiffness is used for deformation ls (= γ l), where l: length from the unloading point to

the intersection of slip line and the line connecting the origin and the negative maximum response

point ( minmin , F D ). The response point during strain softening moves toward the previous maximum

point ( minmin , F D ) or the yield point if no yielding was experienced in the reloading direction.

If unloading takes place during reloading toward previous maximum response point, theunloading stiffness from the previous maximum response point is used. If the response point crosses

s

minF

min D

max D

xo D yp D

eK

sK uK


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33

the zero resistance axis, the response point follows the same slip stiffness previously defined in thereloading direction. The length of slip deformation is defined for l: length from the new unloadingpoint to the intersection of slip line and the line connecting the origin and the maximum response

point ( max max, D F ).

Values for theparameters of this modelrecommended for shearfailing reinforced concretemembers are α =0.4,

β =0.6 and γ =0.95.

Values for flexuredominated members are

α =0.2, β =γ = 0.0.

Park et al. model: Themodel developed by Parket al. (1987) includes (a)stiffness degradation, (b)pinching and (c) strengthdegradation with cycling.The skeleton curve is atrilinear relation. Theextension of unloadingfrom the post-crackingbranch of the virgin loadingcurve intersects thepre-cracking branch of thetrilinear virgin loading inthe direction of unloadingat an ordinate equal toapproximately two timesthe corresponding yieldmoment. The reloadingbranch is initially directedtowards a point on theprevious extremeunloading branch, at amoment ordinate equal toa user-specified

percentage γ (approximately 0.5) of the yieldmoment. Before reaching this point and uponexceedance of the previous maximum permanentdeformation (curvature at the intersection of theprevious extreme unloading branch and thehorizontal axis), the reloading branch stiffens andmoves toward the point of maximum deformation inthe direction of reloading. The strength degrades inproportion to the amount of energy dissipated up tothe current point. The proportionality constantdepends on the amount of longitudinalreinforcement and confining reinforcement.

Origin Oriented Model: Shiga (1976) suggested a

Hysteresis model by Park et al. (1987)

D

F

C

Y

C

Y

Origin-Oriented Model


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simple hysteresis model, in which the response moves on the line connecting the previous absolutemaximum response point and the origin. If the response point reaches the maximum response point,it moves on the skeleton curve. When unloading takes place from a point on the skeleton curve, theresponse point moves on the line connecting the newly attained maximum response point and theorigin.

The model was obtained from the observation on steady-state response of reinforced concretestructural model which oscillated about the origin of the force-deformation relation. No hysteresisenergy is dissipated during the oscillation within theprevious maximum response amplitude. Therefore,viscous damping proportional to the initial stiffness issuggested as a mechanism to dissipate energy withdegradation of stiffness in a system.

Any shape may be used for the skeleton curve ofthis model. This model is sometimes used to representa feature of shear-dominated member, whichdissipates small hysteresis energy and degrades its

stiffness with plastic deformation. The model, however,does not give residual displacement when the loadwas removed. Therefore, the model may not be suitedfor the simulation analysis of response waveform.

Similar to the origin oriented model, the responsepoint may directed toward the previous maximumresponse point on the opposite direction. Such model may be called a peak oriented model.

Matsushima Strength Reduction Model: Short reinforcedconcrete columns, failing in shear, exhibit strength decaywith load reversals and associated stiffness degradation.Matsushima (1969) used a model to explain the damage ofa structure after shear failure in columns. The characteristicsof the model are basically of bilinear type, but the elasticstiffness K n and the yield resistance F n were degradedwhenever unloading takes place from a point on thepost-yielding line in a form;

y

n

n

y

n

n

F F

k K

β

α

=

=

where K y: initial elastic stiffness, F y: initial yield resistance, n:number of unloading from the post-yield stiffness line, α

and β are constants to decay rate.

Sucuoglu’s Energy Based Hysteresis Model: A cycle fatigue model was presented by Sucuogluand Erberik (2004). The model keeps the complete record of energy dissipation and the recordeddissipated energy is used as a memory fluid for determining the amount of stiffness and strengthdeterioration in the subsequent cycle.

The model operates on a bilinear skeleton curve with an initial stiffness oK

and post-yield

stiffness oa K

where a accounts for hardening or softening effects. Pinching is not consideredexplicitly in the general force-deformation reloading paths, however, loss of energy dissipationcapacity due to pinching is the main feature of the model.

Rule 1: the initial elastic region with an initial stiffness oK .

D

F

C

Y

C

Y

Peak-Oriented Model

F

D0k

0

0

N

y y

N

F a F

k b k

=

=

0 yF

k

Matsushima Model


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Rule 2: the post-yield envelope curve has a slope oa K , where a is the post-yield stiffness ratio.

Rule 3: Unloading from the post-yield envelope or from a reloading branch follows a slopeoK until

the entire force in the system is released. If unloading originates from the maximum

displacement point in any direction, then unloading stiffnessu oK K = such as the

unloading branches A1C1, A2C2, A3C3, and A4C4. On the other hand, if unloading originatesfrom an intermediate displacement which is less than the maximum displacement in the

direction, unloading stiffnessuK becomes equal to the slope of the line between the

reloading target B at the current maximum post-elastic displacement and is its unloading

intercept C; e.g., unloading stiffnessuK for A5C5 is equal to the slope of B4C3 and

unloading stiffness uK for A6C6 is equal to the slope of B5C4.

Rule 4: Reloading from an unloading intercept C to a reloading target B follows a sloper K . The

slopes of CiBi are variable and depend on the reduced strength of the target point B at thecurrent maximum displacement in the respective direction. Strength deterioration dependson dissipated energy.

Umemura-Ichinose Modification of Takeda Model: Reinforced concrete members after flexural

yielding exhibit capacity degradation due to cyclic loading especially when subjected to high shear.Umemura et al. (2002) propose to modify the Takeda model to include this capacity degradation.The target point during loading in Takeda model is the previous maximum response point on theskeleton curve without degradation in resistance. Umemura et al. (2002) proposed to use a new

target response point on the skeleton curve at displacementnd larger than that of the previous

maximum response point whenever previous maximum response was exceeded in either direction;

n max min( ) pd d d d χ = + −

where, pd : displacement of the previous target point in the same loading direction, max min,d d : peak

displacements using the previous target point p

d , χ : stiffness degradation factor; which is defined

as

0.12 0.00069 0.039 0.016 0.019 s B w B

L N BD D

χ σ ρ σ = + − + −

Energy-based Hysteresis Model (Sucuoglu, 2004)


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where, B

σ : concrete strength (MPa),w

ρ : lateral reinforcement ratio (%), / B

N BDσ : axial force

ratio, and /s L D : shear span to depth ratio.

The general idea is shown in the figure below.

Umemura et al. modification of Takeda Model (2002)

References:

Banon, H., J. M. Biggs and H. Max Irvine, "Seismic Damage in Reinforced Concrete Frames,"Journal of Structural Division, ASCE, Vol. 107, No. ST9, September 1981, pp. 1713-1729.

Chung, Y. S., et al., “Seismic Damage Assessment of Reinforced Concrete Members,” NationalCenter for Earthquake Engineering Research, State University of New York, Buffalo, TechnicalReport NCEER-87-0022, 1987.

Kato, D., S. Otani, H. Katsumata and H. Aoyama, "Effect of Wall Base Rotation Behavior of

Reinforced Concrete Frame-Wall Building," Proceedings, Third South Pacific RegionalConference on Earthquake Engineering, Victoria University of Wellington, New Zealand, May1983.

Matsushima, Y., "Discussion of Restoring Force Characteristics of Buildings, the Damage fromTokachi-oki Earthquake (in Japanese)," Report, Annual Meeting, Architectural Institute of Japan, August 1969, pp. 587-588.

Park, Y. J., et al., “IDARC: Inelastic Damage Analysis of Reinforced Concrete Frame-Shear WallStructures,” National Center for Earthquake Engineering Research, State University of NewYork at Buffalo, Technical Report NCEER-87-0008, 1987.

Roufaiel, M. S. L., and C. Meyer, "Analytical Modeling of Hysteretic Behavior of R/C Frames,"Journal of Structural Division, ASCE, Vol. 113, No. 3, March 1987, pp. 429-444.

Shiga, T., Vibration of Structures (in Japanese), Structural Series, Vol. 2, Kyoritsu Shuppan, 1976.

Sucuoglu, H., and Atlug Erberik, „Energy-based Hysteresis and Damage Models for DeterioratingSystems,” Earthquake Engineering and Structural Dynamics, No. 33, 2004, pp. 69 - 88.

Takayanagi, T., and W. C. Schnobrich, "Computed Behavior of Reinforced concrete Coupled ShearWalls," Structural Research Series No. 434, Civil Engineering Studies, University of Illinois atUrbana-Champaign, 1976.

Umemura, H., T. Ichinose, K. Ohashi and J. Maekawa, “Development of Restoring ForceCharacteristics for RC Members Considering Capacity Degradation (in Japanese),” Proceedings, Annual Meeting, Japan Concrete Institute, Vol. 24, No. 2, 2002, pp. 1147-1152.


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11.10 Ibara-Medina-Krawinkler Model

The cyclic hysteretic response of a structural member tested in the laboratory indicates that (1)strength deteriorates with the number and amplitude of cycles, even if the displacement associatedwith the strength has not been reached, (2) Strength deterioration occurs after reaching themaximum resistance, (3) Unloading stiffness may also deteriorates, and (4) The reloading stiffness

may deteriorates at an accelerated rate (Ibara, Medina and Krawinkler, 2005).

Backbone Curve: The backbone curve defines the force-deformation relation under monotonicallyincreasing load, defined by initial elastic stiffness K e, yield strength F y, and the strain-hardeningstiffness K s. If deterioration of the backbone curve is included, a softening branch begins at the “cap”

deformationc

δ , which corresponds to the peak strength (F c) of the load-deformation curve. If the

cap deformationc

δ is normalized by the yield deformation, the resulting ratio may be denoted as

ductility capacity ( /c y

δ δ ). The softening branch is defined by the post-capping stiffness,

c c eK K α = , which usually has a negative value. In addition, a residual strength can be assigned to

the model,r y

F F λ = , which represents the fraction of the yield strength of the component that is

preserved once a given deterioration threshold is achieved. The backbone curves can be different inpositive and negative directions in the proposed modeling.

Backbone curve for hysteretic models

The parameters of the backbone curve are normally obtained from experimental results ratherthan theoretical analysis..

Bilinear Model: This model is based on the standard bilinear hysteretic rules with strain hardeningbackbone curve. The strength limit is introduced if the backbone curve includes a branch withnegative slope; i.e., when the response in a direction passes the cap point and in the softeningrange (point 3), response resistance cannot exceeds the smallest strength of the point 3 duringreloading in the direction, for example, after unloading from point 5. The resistance is limited by theresistance at point 3.


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Bilinear Model with Strength Limit

Peak-oriented Model: This model is similar to the Clough Model (Clough and Johnston, 1966)modified by Mahin and Bertero (1976), but the backbone curve is modified to include the strainhardening and softening.

Basic Rules for Peak-oriented Hysteresis Model

Pinching Model: The pinching model is similar to the peak-oriented model, except that reloadingconsists of two parts. Initially the reloading path is directed towards a “break point”, which is afunction of the maximum permanent deformation and the maximum load experienced in the direction

of loading. The break point is defined by the parameters f k , which defines the maximum

2pinched” strength (points 4 and 8), andd k , which defines the displacement of the break point

(points 4’ and 8’). The first part of the reloading branch is defined by,rel a

K and once break point is

reached (points 4’ and 8’), the reloading path is directed towards the maximum deformation of earlier

cycles in the direction of loading (,rel bK ).

If the absolute deformation at reloading (point 13) is larger than the absolute value of

(1 )d per k δ − , the reloading path consists of a single branch that is directed towards the previousdeformation in the direction of loading.


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Pinching Hysteretic Model

(a) Basic Model Rule, (b) Modification if Reloading Deformation is to the Right of Break Point

Ibara, Medina and Krawinkler (2005) suggest to determine cyclic strength and stiffnessdeterioration on the basis of hysteretic energy dissipation.

Some examples of hysteresis relation are shown below;

(a) Basic Strength Deterioration, (b) Post-capping Strength Deterioration, (c) Unloading StiffnessDeterioration, and (d) Acceleration Reloading Stiffness Deterioration

References:Clough, R.W., and S.B. Johnston, “Effect of Stiffness Degradation on Earthquake Ductility

Requirements,” Proceedings, Japan Earthquake Engineering Symposium, Tokyo, Japan, 1966,pp. 227-232.

Ibara, L.F., R. A. Medina, and H. Krawinkler, “Hysteretic Models that Incorporate Strength andStiffness Deterioration,” Earthquake Engineering and Structural Dynamics, Vol. 34, 2005, pp.1489 - 1511.

S.A., and V.V. Bertero, “Nonlinear Seismic Response of a Coupled Wall System,” Journal ofStructural Division, ASCE, Vol. 102, 1976, pp. 1759-1980.


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11.11 Special Purpose Models

Axial Force-Bending Moment Interaction : It isknown that bending resistance varies with existingaxial force in a reinforced concrete section. Theeffect of axial load on flexural yield level wasconsidered by Mahin and Bertero (1976), in whichthe yield moment of the multi-component modelwas varied with the amount of axial load.

Takayanagi and Schnobrich (1976) modifiedthe Takeda model to include the effect of axialforce-bending resistance interaction in the analysisof a coupled structural wall. The skeleton curve istrilinear. A set of trilinear skeleton curves wereprepared for different level of axial force, and thechange in bending resistance with unit axial loadwas evaluated. The moment m is assumed to

vary with curvature φ and axial force n , while

the axial force n is assumed to vary with

curvature φ and axial strain ε ;

( , )

( , )

m m n

n n

φ

φ ε

=

=

The assumption leads to an un-symmetric relation in an incremental form;

( )m m m m n m n

m nn n n

n nn

φ φ ε φ φ φ ε

φ ε φ ε

∂ ∂ ∂ ∂ ∂ ∂ ∂Δ = Δ + Δ = + Δ + Δ

∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂Δ = Δ + Δ∂ ∂

The above relation for incremental curvature φ Δ and strain ε Δ , and then modification factor wasdeveloped to regain the symmetry;

1( ) *

1

1{ } *

1 ( / )( )

mm EI

m n

n m

nn EA

n m m m

n n

φ φ φ

ε ε ε

φ φ

∂Δ = Δ = Δ

∂ Δ∂ −∂ Δ

∂Δ = Δ = Δ

∂ ∂ Δ ∂∂ − −∂ ∂ Δ ∂

where * EI : instantaneous flexural rigidity, and * EA : instantaneous axial rigidity. The ration

mΔΔ

is assumed to remain constant during a small load increment.

The stiffness is updated for the subsequent load increment considering the existing axial forcelevel. For an increase in axial force, the moment-rotation hysteresis relation is directed to thecorresponding loop with increased yield moment.

The axial force-moment interaction effect can be easily handled by "fiber" model. Curvature maybe assumed to distribute uniformly over a specified hinge region, for which a moment-rotationrelation can be evaluated on the basis of the moment-curvature relation at the critical section.

References:

Takayanagi-Schnobrich model foraxial load-moment interaction


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Mahin, S. A., and V. V. Bertero, "Nonlinear Seismic Response of a Coupled Wall System," Journalof Structural Division, ASCE, Vol. 102, 1976, pp. 1759-1780.

Takayanagi, T., and W. C. Schnobrich, "Computed Behavior of Reinforced Concrete Coupled ShearWalls," Structural Research Series No. 434, Civil Engineering Studies, University of Illinois atUrbana-Champaign, 1976.

Hysteresis Model for MS Model: A steel spring and aconcrete spring in the corner of section are located in thesame point, and are subjected to identical displacementhistory. Therefore, the two springs may be combined intoa single composite spring. The skeleton curve isexpressed by a bilinear relation; the compressive yieldresistance is determined as the sum of the compressivestrengths of the concrete and the steel springs, and thetensile yield resistance is equal to the yield resistanceof the steel spring.

Hysteresis relation is of the Takeda model type withthe bilinear skeleton curve; unloading stiffness in acompression zone and in a tension zone was madedifferent:

In a compression zone:

sym

y

mceSE

symcese

D D for D

DK K S

D D for K K S K

>+=

≤+=

−λ

)(

)(

1

11

In a tensile zone:

sym

sy

mse

symse

D D for D

DK S

D D for K S K

−


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The gravity loads was considered as the initial stress.

A response point followed bilinearhysteresis rules between the maximum

response point ( maxmax , F D ) in the tension

side after yielding and a reference point Y'

( y yc F D −, ) on the skeleton curve in the

compression zone. The resistance y

F − at

the reference point was determined at thecompressive yielding of the longitudinalreinforcement.

The unloading stiffness Kr was degradedwith plastic deformation;

α −= )( max

yt

cr

D

DK K

where, yt D : tensile yielding deformation,

max D : maximum deformation greater than

y D , α : unloading stiffness degradation

parameter (= 0.9).

When the response point reached the previous maximum point ( maxmax , F D ) in tension, the

response point moved on the second slope of the skeleton curve, renewing the maximum responsepoint.

When the response point approached the compressive characteristic point Y' (Dyc, -Fy) incompression, the response point was directed to move toward a point Y" (2D yc, -2Fy) from a point P(Dp, Fp) on the bilinear relation:

)( yc x yc p D D D D −+= β

where, β : parameter for stiffness hardening point (=0.2), and x D : deformation at unloading

stiffness changing point. This rule was introduced to reduce an unbalanced force at the compressivecharacteristic point Y' due to a large stiffness change. The compressive characteristic point Y' didnot change under any loading history.

This axial-stiffness hysteresis model was used for the axial deformation of an independentcolumn as well as boundary columns of a wall.

Slip Model: Reinforced concretemembers exhibit slip-type (pinching)behavior before a wide crack closes orwhen longitudinal reinforcing bars slipafter bond deterioration. The slip-typebehavior is characterized by a smallstiffness during reloading at lowresistance level after a large amplitudedeformation in the opposite directionand by the gradual increase in

stiffness with deformation.

Tanabashi and Kaneta (1962) useda slip model with elasto-plastic

Axial force-deformation model forwall boundary element (Kabeyasawa et al., 1983)

Tension

Initial Load

Elongation

Compression


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skeleton curve and zero slip stiffness in their nonlinear response analysis. No hysteresis energy wasdissipated until the response point exceeded the previous maximum response point.

A finite stiffness may be assigned to the slip stiffness and a stress hardening may start to occurbefore the initiation of slip at preceding unloading.

Bond Slip Model: Morita and Kaku(1984) proposed a hysteresis model torepresent the bond stress-bar slip relationon the basis of their observation of thetest results. The model is prepared forassuming various loading situations andmay be useful in a finite element analysisof a reinforced concrete member.

References:

Fillipou, F. C., E. P. Popov and V. V. Bertero, “Effect of Bond Deterioration on Hysteretic Behavior ofReinforced Concrete Joints,” Report No. EERC 83-19, University of California, Berkeley, August1983, 184 pp.

Fillipou, F. C., E. P. Popov and V. V. Bertero, “Modeling of Reinforced Concrete Joints under CyclicExcitations,” Journal, Structural Engineering, ASCE, Vol. 109, No. 11, November 1983, pp.2666 - 2684.

Fillipou, F. C., “A Simplified Model for R

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