1. 1
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 a
section, for a member, for a story and for an entire structure. The resistance-deformation relation of
a structural analysis unit observed in a laboratory test must be idealized into a
resistance-deformation hysteresis model. Different levels of resistance-deformation models must be
used for structural elements considered in an analysis; e.g., a constitutive model of materials in a
finite element method analysis, a hysteresis model for a rotational spring in a one-component
member model, a story shear-drift hysteresis model for a mass-spring model.
A hysteresis model is derived by extracting common features of resistance-deformation relations
observed in laboratory tests of members of similar properties. The hysteresis model of a member
must be able to express resistance-deformation relations under any loading history, including load
reversals.
Resistance-deformation relationship under monotonically increasing loading is called the primary
curve, skeleton curve or backbone curve. The skeleton curve provides an envelope of the hysteresis
resistance-deformation relationship if the behavior is governed by stable flexure. The skeleton curve
for reinforced concrete member is normally represented by a trilinear relation with stiffness changes
at flexural cracking and tensile yielding of longitudinal reinforcement. The skeleton curve of a
member must be defined on the basis of mechanical properties of constitutive materials and
geometry of the member. Some researchers suggest the use of a bilinear relation with a stiffness
change at yielding, ignoring the initial uncracked stage, because a reinforced concrete member
subjected 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, yield
deformation and ultimate deformation. The stiffness degrades from the initial elastic stiffness with
increased inelastic deformation and the number of cycles under reversed loading. The elastic
modulus of concrete varies significantly with concrete strength and mix; initial cracks cause decay in
the stiffness. The estimate of yield deformation is more complicated by the interaction of bending
and shear deformation and additional deformation due to pullout of longitudinal reinforcement from
the anchorage zone and due to bar slip of longitudinal reinforcement along the longitudinal
reinforcement within the member. Empirical expressions are necessary for the estimate of yield and
ultimate deformation.
The coordinates of a response point on a deformation-resistance plane are given by (D, F), in
which, 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 hysteresis
description;
Loading: a case where the absolute value of resistance (or
deformation) 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 the
response point reaching the skeleton curve.
The hysteresis model is formulated on the basis of resistance-deformation relations observed in
the 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

2. 2
amplitude,
(2) Small deflection amplitude
excursion must be placed after a large
amplitude excursion to study the
slip-type behavior
A lateral load-deflection relation of a
reinforced concrete member was
obtained from the test of a slender
column (Otani and Cheung, 1981). The
behavior was dominantly by flexure
although flexural cracks started to
incline due to the presence of high
shear stresses before flexural yielding.
The yielding of the longitudinal
reinforcement 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). This
reduction in stiffness is attributable to the formation of new cracks during loading cycle 3, and also to
a 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 maximum
displacement. Note that the peak-to-peak stiffness of cycle 5 is significantly smaller than that of cycle
2, although the displacement amplitudes of the two cycles are comparable. The peak-to-peak
stiffness 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 the
member dominated by flexural behavior.
A hysteresis model of a reinforced concrete "flexural" member must be able to represent the
above characteristics. The skeleton curve is similar to an "envelope curve" of a force-deformation
relation under load reversals. The state of the art is not sufficient to determine the ultimate point, at
the deformation of which the resistance of a member starts to decay. The force-deformation relation
after the onset of strength decay is normally not modeled because the behavior is strongly
dependent on a particular local deterioration of materials.
If the reinforced concrete is subjected to
high shear stress reversals, or if the
slippage of the reinforcement from concrete
within the anchorage area occurs, the
force-deflection curve exhibits a pronounced
"pinching". The pinching behavior is also
observed;
(a) in a "flexural" member when the
amount of longitudinal reinforcement differs
significantly for the tension and compression
sides at the critical sections, typically in a
girder with monolithically cast slabs,
(b) at a member end where additional
deformation may be caused by anchorage
slip of longitudinal reinforcement within the
adjacent member or connection, and
Hysteresis of slip type (Bertero and Popov, 1977)

3. 3
(c) in a member where bond splitting cracks develop along the longitudinal reinforcement.
Because such hysteresis relationship is highly dependent on loading history and structural
properties of the member, a general hysteresis model is difficult to formulate; or the parameters of
hysteresis models cannot be analytically determined by the properties of the member. In the design
of 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 care
must 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 model
indicates a large memory to store the hysteresis rule program in a computer. It does not lead to a
longer computation time because the complicatedness of a hysteresis model requires simply many
branches in a computer program, and only a few branches are referred to for a step of response
computation.
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 Masing
type models are shown below:
A hysteresis energy dissipation index (Eh) is
used to express the amount of hysteresis energy
dissipation WΔ per cycle during displacement
reversals of equal amplitudes in the positive and
negative directions;
mm
h
DF
W
E
π2
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 WΔ
dissipated by an equivalent viscous damper of a
linearly elastic system in one cycle under the
"resonant" "steady-state" oscillation.
The steady state response amplitude mD
under sinusoidal excitation with amplitude op
and circular frequency ω , is given by
Hysteresis energy dissipation index

4. 4
)sin()(
)(4})(1{
1
222
φω
ω
ω
ω
ω
+=
+−
=
tDtx
h
k
p
D
m
nn
o
m
The energy dissipated WΔ by viscous damper per cycle is
2
2
22
0 0
2
2
)(cos))((
m
m
T T
m
Dmkh
Dc
dttDcdt
dt
dx
dt
dx
cW
n n
ωπ
ωπ
φωω
=
=
+==Δ ∫ ∫
where kcm ,, ,: mass, damping coefficient and stiffness of an SDF system, h : damping factor
(
km
c
2
= ), nT : natural period of the system (
k
m
π2= ), nω : circular frequency of the system
(
m
k
= ).
At the resonant condition ( nωω = ), the energy dissipated per cycle can be expressed
2
2 mDkhW π=Δ
Therefore, the damping factor corresponding to the hysteresis energy dissipation WΔ is
m
m
mmm
D
F
k
DF
W
Dk
W
h
=
Δ
=
Δ
=
ππ 22
2
The equivalent damping factor should not be confused with a damping factor of a viscously
damped 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 Reinforced
Concrete 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 Civil
Engineering, University of Toronto, 1981.
Saatcioglu, M., "Modeling Hysteretic Force-Deformation Relationships for Reinforced Concrete
Elements," ACI-SP127, American Concrete Institute, Detroit, 1991, pp. 153-198.

5. 5
11.2 Bilinear Model
At the initial development stage of nonlinear dynamic analysis, the elastic-perfectly plastic
hysteretic model ("elasto-plastic model") was used by many investigators. The response point
moves on the elastic stiffness line before the yield stress is reached. After yielding, the response
point 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 dissipation
during a small excursion is not included.
A finite positive slope was assigned to
the stiffness after yielding to simulate the
strain hardening characteristics of the
steel and the reinforced concrete
("bilinear model"). Unloading stiffness
after yielding is equal to the initial elastic
stiffness. The stiffness degradation with
inelastic deformation and energy
dissipation during small amplitude
oscillation are not considered in the
model.
Neither the elasto-plastic model nor
the bilinear model represents the
behavior of reinforced concrete and steel
members. The steel member softens
during reloading after plastic deformation
by the "Bauschinger effect." The
response of the elasto-plastic model is
compared with a test result of a
reinforced concrete column above.
When the degradation in stiffness was recognized in the behavior of the reinforced concrete, the
loading and unloading stiffness Kr was proposed to degrade with the previous maximum
displacement (Nielsen and Imbeault, 1970) in a form:
α−
= )(
y
m
yr
D
D
KK
in which, α : unloading stiffness degradation parameters (0 < α <1); Ky: initial elastic stiffness, and
Dm: previously attained maximum displacement in any direction. The unloading stiffness remains
Bilinear Model
Specimen SP-5
Column top displacement, cm
Columntopforce,kN
Response of Bilinear model and RC column

6. 6
constant until the response displacement amplitude exceeds the previous maximum displacement in
either direction. The model is called a "degrading" bilinear hysteresis model." If the value of a is
chosen to be zero, the unloading stiffness does not degrade with yielding. A smaller value of a tends
to yield a larger residual displacement. The degrading bilinear model does not dissipate hysteretic
energy 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 Eh of
the degrading bilinear model is given by
)1)(1(
)}1(){1(2
α
α
βμμββμπ
μββμμβ
−+−
+−−−
=hE
in which β : ratio of the post-yielding stiffness
to the initial elastic stiffness; and μ : "ductility
factor" (ratio of the maximum displacement to
the initial yield displacement).
The equation is valid for a ductility factor
greater than 1.0. The hysteresis energy index of
a regular bilinear model (α = 0) reaches as
high as 0.33 at a ductility factor of 4.0. However,
such large amplitude oscillations do not
continue during an earthquake; no hysteresis
energy is dissipated by the model during small
amplitude oscillations. The total energy
dissipation of the bilinear model over the
duration of an earthquake is much smaller than
that expected from the hysteretic energy
dissipation index.
Reference:
Nielsen, N. N., and F. A. Imbeault, "Validity of
Various Hysteretic Systems," Proceedings,
Third Japan National Conference on
Earthquake Engineering, 1971, pp. 707-714.

7. 7
11.3 Ramberg-Osgood Model
A stress-strain relation of the metal was expressed using three parameters yD , yF and γ by
Ramberg and Osgood (1943), where yD : 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 by
Jennings, is expressed by
)1(
1−
+=
γ
η
yyy 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 (Fy/Dy), and the initial loading curve passes a point (Fy,
(1+η )Dy) 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, Fo), the unloading, load reversal and reloading
branches of the relationship is given by
)
2
1(
22
1−
−
+
−
=
−
γ
η
y
o
y
o
y
o
F
FF
F
FF
D
DD
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. The
resistance F at a given displacement D must be computed numerically, for example, using the
Newton-Rapson's iterative procedure.
The Ramberg-Osgood model is often used for stress-strain relation of the steel in the finite
element analysis or in the lamina model, and for resistance-deformation relation of steel members in
a frame analysis.
The hysteresis energy dissipation index of the Ramberg-Osgood model is expressed as
)1)(
1
2
1(
2
m
m
y
y
h
D
F
F
D
E −
+
−=
γ
η
π
The model can dissipate some hysteresis energy even if the ductility factor is less than unity. The

8. 8
index is sensitive to the exponent γ
of the model, and the hysteresis
energy dissipation capacity increases
with increasing value of the exponent.
References:
Jennings, P. C., "Response of Simple
Yielding Structures to Earthquake
Excitation," Ph.D. Thesis,
California Institute of Technology,
Pasadena, 1963.
Ramberg, W., and W. R. Osgood,
"Description of Stress-Strain
Curves by Three Parameters,"
National Advisory Committee on
Aeronautics, Technical Note 902,
1943.

9. 9
11.4 Degrading Tri-linear Model
A model that simulates dominantly flexural stiffness characteristics of the reinforced concrete was
used extensively in Japan (Fukada, 1969). The primary curve is of tri-linear shape with stiffness
changes at flexural cracking and yielding. Up to yielding, the model behaves in a manner the same
as the bilinear model. When the response exceeds a yield point, response point follows the
strain-hardening part of the
primary curve. Once
unloading takes place from
a point on the primary curve,
the unloading point is
considered to be a new
"yield point" in the direction.
The model behaves in a
bilinear manner between
the positive and negative
"yield points" with stiffness
degraded proportional to
the ratio of the slopes
connecting "current yield
points" and "the initial yield
points."
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 beyond
yielding,
(b) the hysteretic energy dissipation is
large in the first load reversal cycle after
yielding, and becomes steady in the following
cycles, and
(c) the steady hysteretic energy dissipation
is proportional to the displacement amplitude.
The hysteretic energy dissipation index of
the degrading tri-linear model is expressed as
y
c
c
y
h
F
F
K
K
E )1(
2
−=
π
in which Ky: secant stiffness at yielding (=
yy DF / ), and cK : initial elastic stiffness (=
cc DF / ). The index is independent of the
displacement amplitude, but dependent on
the stiffness and resistance ratios at cracking
and yielding. Cracking point of this model
controls the fatness of a hysteresis loop.
Therefore, it is important to choose the
cracking point taking into account the degree
of a hysteresis loop.
Nomura (1976) used an arbitrary skeleton
curve; 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 the
yield point in the direction, similar to the degrading tri-linear model.
Degrading tri-linear model

10. 10
References:
Fukada, Y., "Study on the Restoring Force
Characteristics 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 their
Modeling," Data for Earthquake Resistant
Design for Buildings, No. 65, Magazine of
Architectural Institute of Japan, June 1976.
Nomura model (1976)

11. 11
11.5 Clough Degrading Model
A hysteretic model with an elasto-plastic
skeleton curve was proposed by Clough
and Johnston (1966) to represent the
hysteretic behavior of a reinforced concrete
beam-column sub-assemblage.
During loading, the response point
follows the elasto-plastic skeleton curve.
The unloading stiffness after yielding was
kept equal to the initial elastic stiffness. The
response point during reloading moves
toward the previous maximum response
point in the direction of reloading, simulating
the stiffness degradation. If yielding has not
taken place in the direction of reloading, the
response point moves toward the yield point
in the reloading direction.
A minor deficiency of the Clough model
was pointed out by Mahin and Bertero
(1976). After unloading from point A,
consider a situation in which reloading takes
place from point B. The original Clough
model assumed that the response point
should move toward the previous maximum
response point C. This is not realistic.
Therefore, a minor modification was added
so that the response point should move
toward an immediately preceding unloading
point A during reloading. When the
response point reaches the point A, the
response point moves toward the previous maximum point C.
The model was made more versatile by incorporating the reduction in unloading stiffness Kr with
a maximum displacement in a form:
α−
= )(
y
m
yr
D
D
KK
in which, α : unloading stiffness
degradation parameter; yK : initial elastic
stiffness; and mD : previous maximum
displacement. The different unloading
stiffness may be assigned taking mD to be
a maximum deformation in the direction
unloading takes place.
If the value of a is chosen to be zero, the
unloading stiffness of the model remains
equal to the initial elastic stiffness.
The response of the Clough model is
shown to compare well with the response of
a reinforced concrete column tested in the
structures laboratory.
Clough Model
RC Column
Column Top Displacement, mm
ColumnResistance,kN
-100 -50 0 50 100
100
200
0
-100
-200
D
F
B
C
Y
Y
Kr=Ky
Ky
A
Clough Model
D
F
B
C
Y
Y
Kr
Ky
DmDy
Fy
A
Modified Clough Model

12. 12
Saiidi and Sozen (1979) and Riddell and Newmark (1979) used models similar to the modified
Clough model.
Wang and Shah (1987) introduced the strength and stiffness degradation effect of cumulative
damage. The strength and stiffness degrade in proportion to (1-Dws), where Dws is the Wang and
Shah 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, as
they are defined on the basis of the location of the point of reversal and of the maximum previous
deformation in the direction of loading, on the degraded skeleton curve. The Wang and Shah
damage 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 index
of the modified Clough model is expressed
as
}
)1(
1{
1
μ
μμββ
π
α
+−
−=hE
where β : ratio of post-yielding stiffness to
the initial elastic stiffness, and μ : ductility
factor.
The equation is valid for ductility factor
greater than unity. The Clough model can
continuously dissipate hysteretic energy
even at a small amplitude oscillation after
yielding.
References:
Clough, R. W., and S. B. Johnston, "Effect
of Stiffness Degradation on
Earthquake Ductility Requirements,"
Proceedings, Second Japan National
Conference 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 Systems
subjected 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 of
Reinforced 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 Damage
Concept,” Earthquake Engineering and Structural Dynamics, John Wiley & Sons, Chichester,
Sussex, Vol. 15, 1987, pp. 993 -1003.

13. 13
11.6 Takeda Degrading Model
Based on the experimental observation on the behavior of a number of medium-size reinforced
concrete members tested under lateral load reversals with light to medium amount of axial load, a
hysteresis model was developed by Takeda, Sozen and Nielsen (1970). The model has been widely
used 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 Py.
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 than
Pcr but smaller the yield load Py. The load is then reversed to -P2 such that P2<P1.
Rule: Unload parallel to loading curve for that half cycle.
3. Condition: A load P1 is reached in one direction such that P1 is larger than Pcr but not larger
than the yield load Py. The load is then reversed to -P3 such that P3>P1.
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 primary
curve. If the previous loading cycle contains no such point, go to the preceding cycle and continue
the 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 located
within the horizontal projection of the primary curve for that direction of loading, connect the point at
zero load to the yield point to obtain the loading slope.
5. Condition: The yield load Py is exceeded in one direction.
Rule: Unloading curve follows the slope given by the following equation:
4.0
)(
D
D
kk
y
yr =
in which rk : 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 yD : deflection at yield.
6. Condition: The yield load is exceeded in one direction but the cracking load is not exceeded in
the opposite direction.
Rule: Unloading follows Rule 5. Loading in the other direction continues as an extension of the
unloading 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 the
immediately preceding quarter-cycle crossed the zero-load axis, unload at 70% of the rate based on
rule 2, 3, or 5, whichever governed in the previous loading history, but not at a slope flatter than the
immediately preceding loading slope.
Takeda model included (a) stiffness changes at flexural cracking and yielding, (b) hysteresis rules
for 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. The
unloading stiffness Kr after yielding is given by

14. 14
α−
+
+
=
y
m
yc
yc
r
D
D
DD
FF
K
in which, α : unloading stiffness degradation parameter; and mD : previous maximum displacement
beyond yielding in the direction concerned. The hysteresis rules are extensive and comprehensive.
The hysteretic energy dissipation index of
the Takeda model is expressed as
}
)1(
1
1
1{
1
μ
μββμ
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 Takeda
hysteresis rule was originally developed to
simulate the behavior of reinforced concrete
members. If this model is used to simulate the
behavior of a story or a simplified structure,
some rules need to be simplified.
For example, hysteresis rules prior to
yielding may be simplified such that unloading
takes place toward the origin of the relation
(Muto Model). This model is often used in a
story-based (mass-spring) earthquake
response analysis.

15. 15
Bilinear Takeda Model: The primary curve of the Takeda model can be made bilinear simply
choosing 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 has
more 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 is
sometimes made simple by letting the
response point moves toward the
origin during unloading, and toward
the maximum response point in the
opposite side upon reloading. The
Takeda hysteresis rules are applied
after the yielding.
This model is similar to Clough
Degrading Model, but is more
complicated having rules for inner
hysteresis loops.
Additional modifications of the Takeda model with bilinear backbone curve may be found in
literature (Powell, 1975, Riddle and Newmark, 1979, Saiidi and Sozen, 1979, Saiidi, 1982). Riddle
and Newmark (1979) used a bilinear skeleton curve and unloading stiffness equal to the initial elastic
stiffness; loading occurs either on the strain hardening branch or towards the furthest point attained
in the previous cycle. Saiidi and Sozen (1979) claimed to simplify the Takeda model using a bilinear
skeleton curve; the model, however, is identical to the modified Clough model with reduced
unloading stiffness with maximum deformation, and reloading to the immediate prior unloading point
if 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 Simulated
Earthquakes," Journal, Structural Division, ASCE, Vol. 96, No. ST12, 1970, pp. 2557-2573.
Otani, S., and M. A. Sozen, "Behavior of Multistory Reinforced Concrete Frames During
Earthquakes," Structural Research Series No. 392, Civil Engineering Studies, University of
Illinois, Urbana, 1972.
Powell, G. H., “Supplement to Computer Program DRAIN-2D,” Supplement to Report, DRAIN-2D
User’s Guide, University of California, Berkeley, August 1975.
Riddle, R., and N. M. Newmark, “Statistical Analysis of the Response of Nonlinear Systems
subjected 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)

16. 16
11.7 Pivot Model
Major features of the force-deflection hysteresis results of large-scale reinforced concrete
members are;
(1) Unloading stiffness decreases as displacement
ductility increases,
(2) Following a nonlinear excursion in one direction,
upon load reversal, the force-deflection path crosses the
idealized initial stiffness line prior to reaching the
idealized yield force, and
(3) The effect of pre-cracked stiffness may be
ignored.
The use of the pivot point in defining degraded unloading
stiffness was first proposed by Kunnath et al. (1990).
Four quadrants are defined by the horizontal axis
and 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 and
points P2 and P3 on the negative elastic stiffness line. The resistance of primary pivot points P3 and
P4 is 2 2yFα and the resistance of primary
pivot points P1 and P2 is 1 1yFα .
Pinching Pivot points PP2 and PP4 fix the
degree of pinching following load reversal in
each quadrant. The resistance of pinching
pivot point PP4 is
*
1 1yFβ and the resistance
of pinching pivot point PP2 is
*
2 2yFβ .
(1) The response follows the strength
envelope so long as no displacement
reversal occurs. The strength envelope is
defined by the initial elastic stiffness, yield
resistance Fy, top point T, degradation point
D and failure point F, and can be different in
positive and negative directions.
(2) Once the yield deformation has been
exceeded in either direction, a subsequent
strength 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. The
initial points of S1 and S2 are yield point Y1 and Y2 in each direction. The strength envelope is
defined 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 by
lines joining the pinching pivot point PP4 (PP2) to maximum response point S1 (S2) until the response
point reaches the strength envelope.
(4) The pinching pivot points PP4 and PP2 are initially fixed, but they move toward the
force-deflection origin with the strength degradation. The resistance at a pinching pivot point is given
by
*
i yiFβ where
Q1
Q2
Q3
Q4
4 2 2( )yP Fα
2 1 1( )yP Fα
Y2(Dy2,Fy2)
Y1(Dy1,Fy1)
* *
4 1 1 1 1( , )y yPP D Fβ β
* *
2 2 2 2 2( , )y yPP D Fβ β
Q1
Q2
Q4
Q3
D
F
Y1
Y2
P1
P4 P3
P2
dy dt1 dd1 df1
Fy1
Fy2
dy2
2 2yFα
1 1yFα
dt2dd2df2
PP4
PP2
Ft1
Ft2
D1
T1
F1
T2
D2
F2
*
1 1yFβ
*
2 2yFβ

17. 17
)(
)(
*
*
tiiMAX
ti
iMAX
ii
tiiMAXii
dd
F
F
dd
>=
≤=
ββ
ββ
where iβ defines the degree of pinching
for a ductile flexural response prior to
strength degradation. tiiMAX dd , : maximum
displacement and strength degradation
displacement (displacement at the highest
resistance) 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 point
Si, 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
D2
F2
S1
S2
D1MAX
D2MAX
F2MAX
F1MAX
Ft2
Ft1

18. 18
Modification for Softened Initial Stiffness:
The unlading stiffness of the maximum
displacement excursion in Quadrant Q1 is guided
toward point P1. A new pivot point P1* is defined
on this unloading line at force (1 η+ ) times larger
than the force at point P1. A line extending from
point P1* through origin defines the new softened
elastic loading line K*. Point PP4* is defined by the
intersection of the modified strength envelope
(line between points PP4 and S1) and the new
softened elastic loading line K*.
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.
Kunnath, S. K., A. M. Reinhorn, and Y. J.
Park, “Analytical Modeling of Inelastic
Seismic Response of RC Structures,”
Journal, Structural Engineering Division,
ASCE, Vol. 116, No. 4, April 1990, pp.
996 - 1017,”
Q2Q3
Q4
Q1
S1
D
PP4
PP4*
1 1yFα
1 1(1 ) yFη α+
P1
P1*
F P4
P4*
K*

19. 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 the
strength 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 point
Si, 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 toward
unloading point (D3,F3) followed by Rule 3.
Rule 5: Loading toward maximum response point X(IS) on the same side after Rule 3 passing pivot
point PV2(IS).
Rule 6: Unloading from Rule 5 at point (D5,F5) toward pivot point PV3(IS) on the other side until the
response point crosses displacement axis at displacement D2, or loading toward unloading
point (D5,F5) followed by Rule 5.
Rule 7: Loading toward maximum response point X(IS) after Rule 4 crossing displacement axis at
displacement D4.
Rule 8: Unloading from Rule 7 at point (D5,F5) toward pivot point PV3(IS) on the other side until
response point crosses displacement axis at displacement at D2, or loading toward
unloading 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
D2
Rule 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)

20. 20
Rule 9: Loading initial elastic stiffness after loading on Rule 3 passing pivot point PV2(IS), or
unloading 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 Rule
5 or unloading on the initial elastic stiffness before until the response point crosses the
displacement 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 toward
pivot 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
LV2
LV3
LV4
For each Yi and LVi
(DYi,FYi) and SYi
Rule 2
Rule 2
D2D2
Y4
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
D2
Rule 3Rule 2
Rule 3
D2
Negative Direction
Positive Direction
Rule 1
Rule 2

21. 21
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 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 Direction
Rule 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

22. 22
Rule 5: Loading toward maximum response point X(IS) on the same side after Rule 3
passing pivot point PV2(IS).
Rule 6: Unloading from Rule 5 at point (D5,F5) toward pivot point PV3(IS) on the other
side until the response point crosses displacement axis at displacement D2, or
loading 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

23. 23
Rule 7: Loading toward maximum response point X(IS) after Rule 4 crossing
displacement axis at displacement D4.
Rule 8: Unloading from Rule 7 at point (D5,F5) toward pivot point PV3(IS) on the other
side 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

24. 24
Rule 9: Loading initial elastic stiffness after loading on Rule 3 passing pivot point
PV2(IS), or unloading 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 Rule 5 or unloading on the initial elastic stiffness before until the
response 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
Y1
Y2
LV1
LV2
LV3
LV4
For each Yi and LVi
(DYi,FYi) and SYi
D2
D2
Y4
Negative Direction
Positive Direction
Rule 3
PV2
Rule 9
Rule 9
PV2
Rule 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
Y1
Y2
LV1
LV2
LV3
LV4
For each Yi and LVi
(DYi,FYi,SYi)
D2D2
Y4
Negative Direction
Positive Direction
PV2
Rule 9
PV2
Rule 10
Rule 5

25. 25
11.8 Stable Hysteresis Models with Pinching
The force-deformation relation of a reinforced concrete member is highly dependent on a loading
history, characterized by strength decay with load reversals and pinching behavior at a low stress
level during reloading, when the behavior of the member is dominated by sliding along inclined shear
cracks or slippage of longitudinal reinforcement.
A flexure-dominated reinforced concrete
girder sometimes exhibits a pinching
characteristic when the amount of
longitudinal reinforcement (or bending
resistance) is significantly different at the top
and bottom of section. This is attributable to
the fact that a wide crack in weak side
cannot close due to large residual strain in
tensile reinforcement after load reversal; the
compressive stress must be resisted by the
longitudinal reinforcement before concrete
faces make contact at cracks.
Many hysteresis models have been
developed on the basis of test results of a
particular set of specimens under a specific
loading history. However, the parameters of most models may not be analytically defined by the
member properties (material properties and member geometry).
Takeda-slip Model: Eto and Takeda (1973) modified the Takeda model to incorporate a slip-type
behavior 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 cracking
and yielding levels can be different in positive and negative directions. The performance of the
model is identical to the Takeda model before yielding.
Pinching takes place only when the
yielding has occurred in the direction of
reloading. The reloading (pinching)
stiffness Ks is defined as
m m
s
m o y
F D
K
D D D
γ−
=
−
where oD : displacement at the end of
unloading (resistance equal to zero),
mD and mF : maximum deformation and
resistance in the direction of reloading,
yD : yield deformation in the direction of
reloading, γ : slip stiffness degradation
index (slip stiffness degradation index γ
is suggested to be 0.5). The pinching
stiffness is revised only when the
maximum response point is exceeded in
the direction of reloading.
When the response point crosses a line connecting the origin and the maximum response point in
the direction of reloading, the response point moved toward the previous maximum response point
and then on the skeleton curve. The unloading stiffness is defined in the same manner as the
D
F Y
(Dm,Fm)
Kd
oD
C
C
Y
Ks
Ks’
(Dm’,Fm’)
'oD
Takeda-slip model
Hysteresis Relation of Beams with
Unbalanced Amount of Reinforcement

26. 26
Takeda model.
The same pinching and unloading stiffness is used during reloading and unloading in an inner
loop.
'
'
c y m
d
c y y
F F D
K
D D D
α−
+
=
+
where, 'cF and 'cD : resistance and deformation at cracking on the opposite side, yF and yD :
resistance and deformation at yielding on the unloading side, mD : maximum deformation on the
unloading side, α : unloading degradation index.
Kabeyasawa-Shiohara Model: Kabeyasawa et al. (1983) modified the Takeda-Eto slip model to
represent the behavior of a girder with the amount of longitudinal reinforcement significantly different
at the top and bottom;

27. 27
(1) the pinching occurs only in one direction where the yield resistance is higher than the other
direction,
(2) the pinching occurs only after the initial yielding in the direction of reloading, and
(3) the stiffness Ks during slipping is a function of the maximum response point (Dm, Fm) and the
point of load reversal (Do, Fo=0.0) in the resistance-deformation plane.
The reloading (slip) stiffness Ks, after unloading in the direction of the smaller yield resistance,
was determined as
γ
om
m
om
m
s
DD
D
DD
F
K
−−
=
where ( mm FD , ): deformation and resistance at the previous maximum response point, oD :
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 Kp through the
previous maximum response point (Dm, Fm); the stiffness is reduced from the slope connecting the
origin 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 stiffness
degradation 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 a
single-degree-of-freedom oscillator, including pinching and
strength degradation.
Unloading-reloading loops prior to yielding in either direction
are bilinear, with slopes equal to those of the pre-cracking and
post-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 mD D γ
; i.e.,
( )
ym
s
m o m
DF
K
D D D
γ
=
−
where, mF and mD : resistance and deformation at the
previous maximum response point, and oD : deformation at
load reversal point. Once the response point crosses the line
connecting the origin and the maximum response point, then
response point moves toward the maximum response point.
The unloading stiffness after yielding is reduced from the
elastic stiffness by factor ( / )y rD D α
.
Post-yield strength and stiffness degradation with cycling is
modeled by directing the reloading branch, after modification for
pinching, toward a point at a displacement equal to (1 ) mDλ+
D
F
C
Y
Ks
O
Dc Dy
Fc
Fy
F’c
F’y
D’cD’y

28. 28
and at a moment (1 ) mFλ− , 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 place
parallel 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 Concrete
Frame 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-story
Reinforced 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
Dy
Fc
Fy

29. 29
11.9 Shear-type Hysteresis Models
Reinforced concrete members exhibit progressive loss of strength under reversed cycles of
inelastic deformation due to lack of shear capacity of member or bond resistance along longitudinal
reinforcement; the monotonic strength of such members cannot be attained.
The response of a reinforced concrete member, exhibiting early strength decay, is difficult to
model because such behavior is sensitive to loading history. General features can be summarized
as the decay in resistance with cyclic loading and pinching response during reloading followed by
hardening.
The undesirable features can be avoided or reduced by following design requirements and
detailing of reinforcement. Therefore, hysteresis models for shear-failing performance may not be
necessary for the response analysis of new construction, but may be necessary for the seismic
evaluation 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 coupling
beams connecting parallel structural walls. The skeleton curve is trilinear.
The reloading (loading in the opposite direction after unloading) is made smaller than the stiffness
toward the previous maximum response point in the direction of reloading; the response point moves
toward 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 the
resistance at the previous maximum response point; e.g., the resistance at the target point is
selected on a strength decay guideline which descends from the yield point. After the response
reaches 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 strength
decay is assumed to proportionally increase with the rotation.
Roufaiel-Meyer Model: Roufaiel and Meyer (1987) used a hysteresis model that includes strength
decay, 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

30. 30
The moment resistance of a bilinear
moment-curvature relation was assumed to decay
when a given strain is reached at the extreme
compression fiber. The curvature at the
commencement of strength decay is called the
critical curvature. The degradation in resistance was
assumed to be proportional to the amount by which
the critical curvature was exceeded.
An auxiliary unloading branch AB is drawn
parallel to the elastic branch of the bilinear skeleton
curve until it intersects a line OB through the origin O
parallel to the strain-hardening branch YA of the
skeleton curve. The line connecting this latter point B
of intersection to the point of previous extreme
deformation in the opposite direction defines the end
C of the unloading branch on the horizontal axis. If
yielding has not taken place in the direction of
loading, the yield point is used as the previous
maximum response point.
From that point on reloading is not always directed straight to the point of the previous extreme
post-yield excursion in the direction of reloading, but it may include pinching, depending on the shear
ration, M/Vh. Pinching is accomplished by directing the reloading branch first towards a point on the
elastic branch of the skeleton curve at an ordinate equal to that of the intersection of this branch with
the line of straight reloading to the previous extreme deformation point, times m<1. The second part
of the reloading branch heads towards this latter extreme deformation point. Parameter m assumes
the following values;
for M/Vh<1.5 m=0
for 1.5<M/Vh<4 m=0.4(M/Vh)-0.6
for 4<M/Vh m=1
The slope of slipping stiffness is
'
'
m
s
m o
F
K m
D D
=
−
Chung et al. (1987) extended the
Roufaiel and Meyer model to include
strength and stiffness degradation at
constant amplitude cycling. The
degradation model requires two
additional parameters: the value of
curvature fφ and the moment fm at
failure in monotonic loading. The failure
is defined as rupture or buckling of
longitudinal reinforcement, concrete
crushing, or the reduction of resistance
to 75 %. If the bilinear approximation to
the moment-curvature curve under
monotonic loading is denoted by ( )pm φ ,
a single half-cycle of loading up to
monotonic failure causes a drop in the moment at failure is ( )p f fm mφ − . By extension, during a
half-cycle to a curvature φ the drop in moment relative to the bilinear monotonic envelope ( )pm φ
is given by
Modification of Roufaiel and Meyer model
by Chung (1987)
Y
F A
B
C
D
O sK oD
(Dm,Fm)
Roufaiel and Meyer Model

31. 31
3
2
( ) { ( ) }
y
p f f
f y
m half cycle at m m
φ 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 ( ( ) ,pm mφ φ− Δ ), rather than at ( ( ),pm φ φ ) as in the original Roufaiel
and Meyer model.
Banon-Biggs-Irvine Model: Banon, Biggs and Max
Irvine (1981) modified Takeda hysteresis model by (a)
using a bilinear skeleton curve, (b) incorporating
pinching and stiffness degradation. The pinching
hysteresis was adopted to simulate the propagation of
inclined cracks due to high shear and slippage of
longitudinal reinforcement.
Hysteresis rules are summarized below;
(a) Moment-rotation relationship is elastic up to the
yield point,
(b) Once the yield point is exceeded, loading
proceeds 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 the
bilinear envelope,
2
2
s
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 not
move toward the previous maximum point, but a point on the skeleton curve at deformation greater
than the previous maximum deformation.
* m
m
D
D
α
=
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 a
hysteresis model to represent the behavior of a
reinforced concrete member failing in shear, in
which strength decay and stiffness reduction due to
load reversals were incorporated. A trilinear skeleton
curve was used with stiffness changes at A and B.
By choosing the skeleton curve without descending
stiffness, the stable flexural behavior may be
represented by this model. The trilinear skeleton
curve may include descending slopes. The following
example shows a skeleton curve with two
descending slopes.
F
D
Y
Y’
Dm
D’m
sK
1K
2K
Banon-Biggs-Irvine Model (1981)
D
F
A
B
Skeleton Curves of Kato Model (1983)

32. 32
The response is linearly elastic before the response point reaches point A. The response point
follows the skeleton curve if the slope of the skeleton curve is positive; if the slope of the skeleton
curve 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 maximum
response 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, Fmax). Then the response point follows a line with
reduced stiffness uK ;
max
( )u e
y
D
K K
D
α−
=
where α : unloading stiffness degradation index, yD : yield deformation.
Upon reloading after crossing zero resistance line, the response point moves on a line with
reloading (slip) stiffness sK ;
maxmin
min
( )s
o y
DF
K
D D D
β−
=
−
where ( minmin , FD ): previous maximum response point on the skeleton curve in the direction of
reloading, oD : deformation at the completion of unloading, yD : 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 , FD ). The response point during strain softening moves toward the previous maximum
point ( minmin , FD ) or the yield point if no yielding was experienced in the reloading direction.
If unloading takes place during reloading toward previous maximum response point, the
unloading stiffness from the previous maximum response point is used. If the response point crosses
s
minF
minD
maxD
xoD
ypD
eK
sK uK

33. 33
the zero resistance axis, the response point follows the same slip stiffness previously defined in the
reloading direction. The length of slip deformation is defined for l: length from the new unloading
point to the intersection of slip line and the line connecting the origin and the maximum response
point ( max max,D F ).
Values for the
parameters of this model
recommended for shear
failing reinforced concrete
members are α =0.4,
β =0.6 and γ =0.95.
Values for flexure
dominated members are
α =0.2, β =γ = 0.0.
Park et al. model: The
model developed by Park
et al. (1987) includes (a)
stiffness degradation, (b)
pinching and (c) strength
degradation with cycling.
The skeleton curve is a
trilinear relation. The
extension of unloading
from the post-cracking
branch of the virgin loading
curve intersects the
pre-cracking branch of the
trilinear virgin loading in
the direction of unloading
at an ordinate equal to
approximately two times
the corresponding yield
moment. The reloading
branch is initially directed
towards a point on the
previous extreme
unloading branch, at a
moment ordinate equal to
a user-specified
percentage γ (approximately 0.5) of the yield
moment. Before reaching this point and upon
exceedance of the previous maximum permanent
deformation (curvature at the intersection of the
previous extreme unloading branch and the
horizontal axis), the reloading branch stiffens and
moves toward the point of maximum deformation in
the direction of reloading. The strength degrades in
proportion to the amount of energy dissipated up to
the current point. The proportionality constant
depends on the amount of longitudinal
reinforcement 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

34. 34
simple hysteresis model, in which the response moves on the line connecting the previous absolute
maximum 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, the
response point moves on the line connecting the newly attained maximum response point and the
origin.
The model was obtained from the observation on steady-state response of reinforced concrete
structural model which oscillated about the origin of the force-deformation relation. No hysteresis
energy is dissipated during the oscillation within the
previous maximum response amplitude. Therefore,
viscous damping proportional to the initial stiffness is
suggested as a mechanism to dissipate energy with
degradation of stiffness in a system.
Any shape may be used for the skeleton curve of
this model. This model is sometimes used to represent
a feature of shear-dominated member, which
dissipates small hysteresis energy and degrades its
stiffness with plastic deformation. The model, however,
does not give residual displacement when the load
was removed. Therefore, the model may not be suited
for the simulation analysis of response waveform.
Similar to the origin oriented model, the response
point may directed toward the previous maximum
response point on the opposite direction. Such model may be called a peak oriented model.
Matsushima Strength Reduction Model: Short reinforced
concrete columns, failing in shear, exhibit strength decay
with load reversals and associated stiffness degradation.
Matsushima (1969) used a model to explain the damage of
a structure after shear failure in columns. The characteristics
of the model are basically of bilinear type, but the elastic
stiffness Kn and the yield resistance Fn were degraded
whenever unloading takes place from a point on the
post-yielding line in a form;
y
n
n
y
n
n
FF
kK
β
α
=
=
where Ky: initial elastic stiffness, Fy: 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 Sucuoglu
and Erberik (2004). The model keeps the complete record of energy dissipation and the recorded
dissipated energy is used as a memory fluid for determining the amount of stiffness and strength
deterioration 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 considered
explicitly in the general force-deformation reloading paths, however, loss of energy dissipation
capacity 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
D
0k
0
0
N
y y
N
F a F
k b k
=
=
0yF
k
Matsushima Model

35. 35
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 slope oK until
the entire force in the system is released. If unloading originates from the maximum
displacement point in any direction, then unloading stiffness u oK K= such as the
unloading branches A1C1, A2C2, A3C3, and A4C4. On the other hand, if unloading originates
from an intermediate displacement which is less than the maximum displacement in the
direction, unloading stiffness uK 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 stiffness uK 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 slope rK . The
slopes of CiBi are variable and depend on the reduced strength of the target point B at the
current maximum displacement in the respective direction. Strength deterioration depends
on 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 the
skeleton curve without degradation in resistance. Umemura et al. (2002) proposed to use a new
target response point on the skeleton curve at displacement nd 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 pd , χ : stiffness degradation factor; which is defined
as
0.12 0.00069 0.039 0.016 0.019 s
B w
B
LN
BD D
χ σ ρ
σ
= + − + −
Energy-based Hysteresis Model (Sucuoglu, 2004)

36. 36
where, Bσ : concrete strength (MPa), wρ : lateral reinforcement ratio (%), / BN BDσ : axial force
ratio, and /sL 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,” National
Center for Earthquake Engineering Research, State University of New York, Buffalo, Technical
Report 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 Regional
Conference on Earthquake Engineering, Victoria University of Wellington, New Zealand, May
1983.
Matsushima, Y., "Discussion of Restoring Force Characteristics of Buildings, the Damage from
Tokachi-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 Wall
Structures,” National Center for Earthquake Engineering Research, State University of New
York 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 Deteriorating
Systems,” Earthquake Engineering and Structural Dynamics, No. 33, 2004, pp. 69 - 88.
Takayanagi, T., and W. C. Schnobrich, "Computed Behavior of Reinforced concrete Coupled Shear
Walls," Structural Research Series No. 434, Civil Engineering Studies, University of Illinois at
Urbana-Champaign, 1976.
Umemura, H., T. Ichinose, K. Ohashi and J. Maekawa, “Development of Restoring Force
Characteristics for RC Members Considering Capacity Degradation (in Japanese),” Proceedings,
Annual Meeting, Japan Concrete Institute, Vol. 24, No. 2, 2002, pp. 1147-1152.

37. 37
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 associated
with the strength has not been reached, (2) Strength deterioration occurs after reaching the
maximum 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 monotonically
increasing load, defined by initial elastic stiffness Ke, yield strength Fy, and the strain-hardening
stiffness Ks. If deterioration of the backbone curve is included, a softening branch begins at the “cap”
deformation cδ , which corresponds to the peak strength (Fc) of the load-deformation curve. If the
cap deformation cδ is normalized by the yield deformation, the resulting ratio may be denoted as
ductility capacity ( /c yδ δ ). The softening branch is defined by the post-capping stiffness,
c c eK Kα= , which usually has a negative value. In addition, a residual strength can be assigned to
the model, r yF Fλ= , 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 in
positive and negative directions in the proposed modeling.
Backbone curve for hysteretic models
The parameters of the backbone curve are normally obtained from experimental results rather
than theoretical analysis..
Bilinear Model: This model is based on the standard bilinear hysteretic rules with strain hardening
backbone curve. The strength limit is introduced if the backbone curve includes a branch with
negative slope; i.e., when the response in a direction passes the cap point and in the softening
range (point 3), response resistance cannot exceeds the smallest strength of the point 3 during
reloading in the direction, for example, after unloading from point 5. The resistance is limited by the
resistance at point 3.

38. 38
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 strain
hardening and softening.
Basic Rules for Peak-oriented Hysteresis Model
Pinching Model: The pinching model is similar to the peak-oriented model, except that reloading
consists of two parts. Initially the reloading path is directed towards a “break point”, which is a
function of the maximum permanent deformation and the maximum load experienced in the direction
of loading. The break point is defined by the parameters fk , which defines the maximum
2pinched” strength (points 4 and 8), and dk , which defines the displacement of the break point
(points 4’ and 8’). The first part of the reloading branch is defined by ,rel aK 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 perk δ− , the reloading path consists of a single branch that is directed towards the previous
deformation in the direction of loading.

39. 39
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 stiffness
deterioration 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 Stiffness
Deterioration, 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 and
Stiffness 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 of
Structural Division, ASCE, Vol. 102, 1976, pp. 1759-1980.

40. 40
11.11 Special Purpose Models
Axial Force-Bending Moment Interaction: It is
known that bending resistance varies with existing
axial force in a reinforced concrete section. The
effect of axial load on flexural yield level was
considered by Mahin and Bertero (1976), in which
the yield moment of the multi-component model
was varied with the amount of axial load.
Takayanagi and Schnobrich (1976) modified
the Takeda model to include the effect of axial
force-bending resistance interaction in the analysis
of a coupled structural wall. The skeleton curve is
trilinear. A set of trilinear skeleton curves were
prepared for different level of axial force, and the
change in bending resistance with unit axial load
was 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 n
n n n
n n
n
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 φΔ and strain εΔ , and then modification factor was
developed to regain the symmetry;
1
( ) *
1
1
{ } *
1 ( / )( )
m
m EI
m n
n m
n
n 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 ratio
n
m
Δ
Δ
is assumed to remain constant during a small load increment.
The stiffness is updated for the subsequent load increment considering the existing axial force
level. For an increase in axial force, the moment-rotation hysteresis relation is directed to the
corresponding loop with increased yield moment.
The axial force-moment interaction effect can be easily handled by "fiber" model. Curvature may
be assumed to distribute uniformly over a specified hinge region, for which a moment-rotation
relation can be evaluated on the basis of the moment-curvature relation at the critical section.
References:
Takayanagi-Schnobrich model for
axial load-moment interaction

41. 41
Mahin, S. A., and V. V. Bertero, "Nonlinear Seismic Response of a Coupled Wall System," Journal
of Structural Division, ASCE, Vol. 102, 1976, pp. 1759-1780.
Takayanagi, T., and W. C. Schnobrich, "Computed Behavior of Reinforced Concrete Coupled Shear
Walls," Structural Research Series No. 434, Civil Engineering Studies, University of Illinois at
Urbana-Champaign, 1976.
Hysteresis Model for MS Model: A steel spring and a
concrete spring in the corner of section are located in the
same point, and are subjected to identical displacement
history. Therefore, the two springs may be combined into
a single composite spring. The skeleton curve is
expressed by a bilinear relation; the compressive yield
resistance is determined as the sum of the compressive
strengths of the concrete and the steel springs, and the
tensile yield resistance is equal to the yield resistance
of the steel spring.
Hysteresis relation is of the Takeda model type with
the bilinear skeleton curve; unloading stiffness in a
compression zone and in a tension zone was made
different:
In a compression zone:
sym
y
m
ceSE
symcese
DDfor
D
D
KKS
DDforKKSK
>+=
≤+=
−λ
)(
)(
1
11
In a tensile zone:
sym
sy
m
se
symse
DDfor
D
D
KS
DDforKSK
−<=
−≥=
−
'
'
'
2
22
λ
where Kse and Kce: initial elastic stiffness of the steel spring and the concrete spring, Dsy: yield
deformation of the concrete and steel springs, Dm: previous maximum response deformation in
compression, Dm': previous maximum deformation in tension, S1 = S2 =2.0 and k = 0.4.
Post yielding stiffness was chosen to be 0.02 times the initial elastic stiffness of the direction of
loading. Upon reloading in compression, the response point moves on the slip stiffness line toward a
point (Dm, Fm"), where mm FF θ=" and θ = 0.4. When the sign of deformation changes, the
response point moves toward the previous maximum point in compression. Similar to the Takeda
model, the response point moves toward a peak of immediately outer loop.
Axial Force-deformation Model: Kabeyasawa and Shiohara et al. (1983) used a hysteresis model
for an axial force-deformation relation of a boundary column in the analysis of a structural wall. The
model was developed on the basis of the observed axial deformation behavior of the boundary
column in the test of the full-scale seven-story structure tested as a part of U.S.-Japan Cooperative
Program (Yoshimura and Kurose, 1985).
The tension stiffening was ignored; concrete was assumed to resist no tensile stress. The axial
stiffness in tension was made equal to the stiffness of the reinforcing steel in the boundary column,
and the stiffness in compression was assumed to be linearly elastic including the stiffness of the
concrete. The stiffness in tension changed at the tensile yielding of the longitudinal reinforcement.
Composite spring hysteresis model
for MS model
Tension Compression

42. 42
The gravity loads was considered as the initial stress.
A response point followed bilinear
hysteresis rules between the maximum
response point ( maxmax , FD ) in the tension
side after yielding and a reference point Y'
( yyc FD −, ) on the skeleton curve in the
compression zone. The resistance yF− at
the reference point was determined at the
compressive yielding of the longitudinal
reinforcement.
The unloading stiffness Kr was degraded
with plastic deformation;
α−
= )( max
yt
cr
D
D
KK
where, ytD : tensile yielding deformation,
maxD : maximum deformation greater than
yD , α : unloading stiffness degradation
parameter (= 0.9).
When the response point reached the previous maximum point ( maxmax , FD ) in tension, the
response point moved on the second slope of the skeleton curve, renewing the maximum response
point.
When the response point approached the compressive characteristic point Y' (Dyc, -Fy) in
compression, the response point was directed to move toward a point Y" (2Dyc, -2Fy) from a point P
(Dp, Fp) on the bilinear relation:
)( ycxycp DDDD −+= β
where, β : parameter for stiffness hardening point (=0.2), and xD : deformation at unloading
stiffness changing point. This rule was introduced to reduce an unbalanced force at the compressive
characteristic point Y' due to a large stiffness change. The compressive characteristic point Y' did
not change under any loading history.
This axial-stiffness hysteresis model was used for the axial deformation of an independent
column as well as boundary columns of a wall.
Slip Model: Reinforced concrete
members exhibit slip-type (pinching)
behavior before a wide crack closes or
when longitudinal reinforcing bars slip
after bond deterioration. The slip-type
behavior is characterized by a small
stiffness during reloading at low
resistance level after a large amplitude
deformation in the opposite direction
and by the gradual increase in
stiffness with deformation.
Tanabashi and Kaneta (1962) used
a slip model with elasto-plastic
Axial force-deformation model for
wall boundary element (Kabeyasawa et al., 1983)
Tension
Initial Load
Elongation
Compression

43. 43
skeleton curve and zero slip stiffness in their nonlinear response analysis. No hysteresis energy was
dissipated 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 occur
before the initiation of slip at preceding unloading.
Bond Slip Model: Morita and Kaku
(1984) proposed a hysteresis model to
represent the bond stress-bar slip relation
on the basis of their observation of the
test results. The model is prepared for
assuming various loading situations and
may be useful in a finite element analysis
of a reinforced concrete member.
References:
Fillipou, F. C., E. P. Popov and V. V. Bertero, “Effect of Bond Deterioration on Hysteretic Behavior of
Reinforced Concrete Joints,” Report No. EERC 83-19, University of California, Berkeley, August
1983, 184 pp.
Fillipou, F. C., E. P. Popov and V. V. Bertero, “Modeling of Reinforced Concrete Joints under Cyclic
Excitations,” Journal, Structural Engineering, ASCE, Vol. 109, No. 11, November 1983, pp.
2666 - 2684.
Fillipou, F. C., “A Simplified Model for Reinforcing Bar Anchorages under Cyclic Excitations,” Report
No. EERC 85-05, University of California, Berkeley, March 1985, 61 pp.
Kabeyasawa, T., H. Shiohara, S. Otani and H. Aoyama, "Analysis of the Full-scale Seven-story
Reinforced Concrete Test Structure," Journal, Faculty of Engineering, University of Tokyo (B),
Vol. XXXVII, No. 2, 1983, pp. 432-478.
(a) Reloading relation (S > SA)
Bond Stress
Bond Stress
Slip Slip
Bond Stress
Slip

44. 44
Li, K.-N., S. Otani and H. Aoyama, "Study on the Elastic-plastic Behavior of Reinforced Concrete
Columns subjected to Bi-directional Horizontal Earthquake Forces and Varying Axial Load (in
Japanese)," Report, Aoyama Laboratory, Department of Architecture, Faculty of Engineering,
University of Tokyo, March 1990.
Morita, S., and T. Kaku, "Slippage of Reinforcement in Beam-column Joint of Reinforced Concrete
Frames," Proceedings, Eighth World Conference on Earthquake Engineering, San Francisco, U.
S. A., Vol. 6, 1984, pp. 477-484.
Tanabashi, R., and K. Kaneta, "On the Relation between the Restoring Force Characteristics of
Structures and the Pattern of Earthquake Ground Motion," Proceedings, Japan National
Conference on Earthquake Engineering, November 1962, pp. 57-62.
Yoshimura, M., and Y. Kurose, "Inelastic Behavior of the Building," ACI SP-84, Earthquake Effects
on Reinforced Concrete Structures, U.S.-Japan Research, American Concrete Institute, Detroit,
1985, pp. 163-202.

45. 45
11.12 Hysteresis Model for Prestressed Concrete Members
This note summarizes a hysteresis model for prestressed concrete members. The model was
proposed by M. Hayashi et al. (1995).
Member end moment M and rotation
θ of a prestressed concrete member under
anti-symmetric bending moment distribution
with the inflection point at mid span is
considered. The skeleton curve of the
moment-curvature ( , Mθ ) relation is
represented by a trilinear relation with
stiffness changes at flexural cracking of
concrete and tensile yielding of longitudinal
reinforcement.
Trilinear Skeleton Relation: The two points to define a trilinear skeleton curve may be estimated as
follows.
(1) Initial elastic stiffness 1K is calculated for a prismatic line member considering flexural and
shear deformation:
1 2
3 c c c c
L
K
L
E I G A
κ
=
+
where L : member length from the face of the orthogonal member to the inflection point (0ne-half of
clear span or height), cE and cG : elastic and shear moduli of concrete, eI : moment of inertia of
the transformed concrete section, cA : cross sectional area of the transformed concrete section, κ :
shape factor for shear deformation (=1.2 for a rectangular section).
(2) Cracking moment cM and rotation cθ are calculated for a condition that the tensile stress at
the extreme tensile fiber reaches the tensile strength tσ of concrete:
1
( )e
c t e
c
c
c
P
M Z
A
M
K
σ
θ
= +
=
where, eP : axial force acting on the section including effective prestressing force, cA : cross
sectional area of concrete, eZ : section modulus of the transformed section. Tensile strength tσ of
concrete may be assumed to be equal to 1.8 Bσ in kgf/cm2
where compressive strength Bσ of
concrete is expressed in kgf/cm2
.
(3) Yielding moment yM should be calculated for a given axial force and effective prestressing
force assuming (a) plain section to remain plain after bending, (b) nonlinear axial stress-strain
relation of concrete and reinforcement, and (c) equilibrium of internal and external forces. A parabola
and straight descending line may be used to represent stress-strain relation of concrete in
compression; an elasto-plastic stress-strain relation may be used for steel reinforcement ignoring
strain hardening.
Rotation yθ at yielding may be evaluated by integrating the curvature along the member, but
this often underestimates the deformation. Sugano (1970) proposed an empirical expression for the
Am
B Am m=
δ
Aθ
B Aθ θ=

46. 46
ratio of secant stiffness at yielding to the initial elastic stiffness as follows;
2
1
{0.43 1.64 0.043 0.33 }( )
y
y
y t
B
M
a N d
n p
K D b D D
θ
α
σ
= = + + +
where n : modular ratio of steel to concrete, tp :
tensile reinforcement ratio including prestressing
reinforcement area as increased by the yield stress
ratio of prestressing reinforcement to ordinary
reinforcement, a : shear span, b and D : width and
depth of member section, N : axial force of section
including effective prestressing force, d : effective
depth of section, Bσ : compressive strength of
concrete.
Characteristic Points on Hysteresis Relations:
The following points and stiffness are used in this
model.
(1) Characteristic point A ( ,A AMθ ) is defined on the
initial elastic stiffness line with stiffness 1K . This
point is used for the hysteresis relation of a prestressed concrete member. Moment resistance AM
of the characteristic point is defined as decompression moment; i.e., for the effective prestressing
force Pe,
e
A e
c
P
M Z
A
=
The moment is zero for a reinforced concrete member without prestressing force. Rotation Aθ is
calculated for moment AM and initial stiffness 1K ;
1
A
A
M
K
θ =
(2) Characteristic point B ( ,B BMθ ) is defined for Takeda hysteresis model (Takeda, Sozen and
Nielsen, 1970) as the terminal point (zero moment resistance) of unloading from the maximum
response point M ( ,m mMθ ). The unloading stiffness BK is defined as follows;
(a) unloading before yielding:
'
'
m c
B
m c
M M
K
θ θ
−
=
−
(b) unloading after yielding:
'
( )
'
y c m
B
y c y
M M
K γθ
θ θ θ
−
−
=
−
where yM and yθ : yield moment and rotation on the side of the unloading point, 'cM and 'cθ :
cracking moment and rotation on the opposite side, and γ : unloading stiffness degradation index of
the Takeda model (=0.5 for normal reinforced concrete members). The rotation Bθ is calculated as
m
B m
B
M
K
θ θ= −
C
Y
yθcθ
cM
yM
Rotation
Moment
1K
1y Kα
A
Aθ
AM

47. 47
C
Y
Rotation
Moment
A
C’
'cM
'cθ
C
Y
yθ
yM
Rotation
Moment
A
C’
'cM
'cθ
M
mM
mθ
B
Bθ
BK
B
BK
mθBθ
M
(3) Unloading stiffness AK of fully prestressed concrete members from the maximum response
point M ( ,m mMθ ) on the skeleton curve is defined as follows;
C
Y
Rotation
Moment
A
Aθ
AM
M
mM
AK
C
Y
yθ
yM
Rotation
Moment
A
Aθ
AM
MmM
mθ
AK
mθ
(a) unloading before yielding:
m A
A
m A
M M
K
θ θ
−
=
−
(b) unloading after yielding:
( )
y A m
A
y A y
M M
K γθ
θ θ θ
−
−
=
−
(4) Unloading stiffness DK of this model from the maximum response point M ( ,m mMθ ) on the
skeleton curve is defined by index 'λ taking into consideration the characteristics of both reinforced
concrete and fully prestressed concrete members.
' (1 ')D A BK K Kλ λ= + −
(5) Characteristic point D ( ,D DMθ ) is defined as an intersection of line AB and the unloading line
MD of this model with unloading stiffness DK from the maximum response point M( ,m mMθ ) on the

48. 48
skeleton curve.
Coordinates of the intersection D ( ,D DMθ ) must satisfy the two equations;
:
:
A D
A B D B
m D
D
m D
M M
line AB
M M
K line MD
θ θ θ θ
θ θ
=
− −
−
=
−
The moment resistance DM is thus solved from the two simultaneous equations as
m
m B
D
D A
A
A B
D
M
K
M M
M
K
θ θ
θ θ
− −
=
− −
(6) Unloading stiffness EK after reaching characteristic point D
(6-1) No yielding has taken place on the unloading side:
1EK K= (initial elastic stiffness)
(a-1) The terminal point of this unloading stiffness is point E’ at moment level equal to moment
'AM of characteristic point A’ on the opposite side if no cracking has taken place on the
opposite side.
(a-2) The terminal point of this unloading stiffness is point E’ at moment level equal to moment
'DM of characteristic point D’, which was defined during unloading after cracking on the
other side.
(6-2) Yielding has already occurred on the unloading side,
1 1
1
' ( )
'
( )
'
' ' (1 ')
m
y
y c m
B
y c y
E B
K K
M M
K
K K K
γ
γ
θ
θ
θ
θ θ θ
λ λ
−
−
=
−
=
−
= + −
where rotation mθ is the maximum response rotation where the unloading initiated. The terminal
point of this unloading stiffness is point E’ at moment level equal to moment 'DM of
characteristic point D’ in the opposite direction.
DK
Rotation
Moment
Y
M
C
A
D
B
mM
AM
mθBθAθ
Rotation
Moment
C
Y
M
DK
A
D
B
mM
AM
Aθ yθ mθ

49. 49
Characteristics of Prestressed Concrete:
(1) Unlading stiffness degradation index γ
Index γ was initially used in
the Takeda hysteresis model
(Takeda, Sozen and Nielsen,
1970) to control the unloading
stiffness from the maximum
deformation on the post-yielding
skeleton curve and also to control
the hysteresis area per cycle. The
unloading stiffness was degraded
from the reference stiffness,
defined as a slope connecting the
yield point of unloading side and
the cracking point on the opposite
side.
'
( )
'
y c m
B
y c y
M M
K γθ
θ θ θ
−
−
=
−
The index value from 0.4 to 0.5 is normally used for reinforced concrete members. The hysteresis
energy dissipation decreases with increasing value of the index.
The unloading stiffness in moment-rotation relations of reinforced concrete and prestressed
concrete member tests were examined. The following graph was suggested by Hayashi et al. (1995)
to express the unloading stiffness degradation index γ as a function of the ratio λ of ultimate
moment resistance attributable to prestressing reinforcement to the total ultimate moment. The ratio
λ is zero for a reinforced concrete member, unity for a fully prestressed concrete member, and
between zero and unity for a partially prestressed concrete member. The unloading stiffness
degradation index γ for partially prestressed to fully prestressed concrete is 0.7 to 0.8. The
hysteresis energy dissipation per cycle is less compared with a reinforced concrete member.
(2) Index 'λ for unloading stiffness
Index 'λ controls unloading
stiffness of reinforced concrete,
partially prestressed reinforced
concrete and fully prestressed
concrete members. The unloading
stiffness of the moment-rotation
relations of member tests was
examined. Hayashi et al. (1995)
suggested expressing index 'λ
for unloading stiffness as a function
of the ratio λ of ultimate moment
resistance attributable to
prestressing reinforcement to the
total ultimate moment. The index
'λ increases with increasing
ultimate moment ratio λ .
The effect of prestressing on unloading stiffness is negligible when the ultimate moment ratio was
less than 0.3.
0.0 0.2 0.4 0.6 0.8 1.0
1.0
0.0
0.2
0.4
0.6
0.8
Unloadingstiffnessdegradationindex
Ultimate moment ratio of prestressing reinforcementλ
Indexforunloadingstiffness
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Ultimate moment ratio λ of prestressing reinforcement

50. 50
Hysteresis Rules:
Rule 1: Before flexural cracking at C ( ,c cMθ ), the
relation is linearly elastic with stiffness 1K .
If the response point reaches flexural cracking point C
( ,c cMθ ), the response point follows Rule 2.
Rule 2: The response point ( , Mθ ) moves on the
second skeleton line CY toward yield point Y ( ,y yMθ )
with stiffness 2K .
If the response point reaches the yielding point Y
( ,y yMθ ), the response point follows Rule 3.
If the unloading takes place from the maximum
response point M ( ,m mMθ ), the response point follows Sub-rule 2-1.
Sub-rule 2-1: The response point moves on line MD
with unloading stiffness DK , where M ( ,m mMθ ) is the
previous maximum point on the second skeleton line
CY. The unloading stiffness DK is defined as
'
'
' (1 ')
m A
A
m A
m c
B
m c
D A B
M M
K
M M
K
K K K
θ θ
θ θ
λ λ
−
=
−
−
=
−
= + −
Characteristic point D is defined as the intersection of
the unloading line MD and line AB connecting two
characteristic points A and B. The moment DM at
point D is given by
m
m B
D
D A
A
A B
D
M
K
M M
M
K
θ θ
θ θ
− −
=
− −
The response point moves on line MD during unloading and reloading.
If the response point reaches the previous maximum point M ( ,m mMθ ) during reloading, the
response point follows Rule 2.
If the response point reaches point D during unloading, the response point follows Sub-rule
2-2.
Sub-rule 2-2: The response point follows Sub-rule 2-2-1 if no cracking has taken place in the
opposite direction, or Sub-rule 2-2-2 if cracking has taken place in the opposite direction.
Sub-rule 2-2-1: The response point moves elastically from the characteristic point D
( ,D DMθ ) of Sub-rule 2-1 to point E’ whose moment level is equal to moment 'AM of
C
Y
yθcθ
cM
yM
Rotation
Moment
1K
2K
Rule 1
Rule 2
Rule 3
Moment-rotation relation during loading
C
Rotation
A
AM
M
mθ
B
DK
Y
D
Moment
Bθ
mM
Aθ

51. 51
characteristic point A’ in the direction of reloading.
The slope EK of line DE’ is equal to the initial
stiffness 1K . Between characteristic point D and
cracking point E’, the response point moves on line
DE’ during reloading and unloading.
The previous response point M’ on the side of
point E’ is defined as the cracking point C’. The
unloading stiffness 'DK from the previous
maximum M’ (cracking point C’) is defined as 1K .
Characteristic point D’ is defined as the
characteristic point A’, and moment 'DM of the
characteristic point D’ is equal to moment 'AM .
If the response point reaches point E’, the
response point follows Rule 4.
Sub-rule 2-2-2: The response point moves
elastically from the characteristic point D ( ,D DMθ )
of Sub-rule 2-1 to point E’ whose moment level is
equal to moment level 'DM of characteristic point
D’ in the direction of reloading. The unloading
stiffness EK is equal to the initial elastic stiffness 1K . Point D’ and its moment 'DM have
been defined by Sub-rule 2-1 or
Sub-rule 3-1 upon previous unloading
from point M’ ( ' , 'm mMθ ) on the second
skeleton line.
Between characteristic points D and
E’, the response point moves on the
same line DE’ during reloading and
unloading.
If the response point reaches
characteristic point D during reloading,
the response point moves toward
previous maximum response point M
( ,m mMθ ) in the direction of reloading
following Sub-rule 2-1.
If the response point reaches point E’
after crossing zero moment axis
(moment reversal), the response point
follows Sub-rule 4.
Rule 3: The response point follows the third
skeleton line with stiffness 3K .
If the unloading takes place at maximum
response point M ( ,m mMθ ), the response point
follows Sub-rule 3-1.
1K
A
B
C
D
E’A’
C’
1EK K=
mθ
mM
No cracking in
reloading
direction
M
DK
Sub-rule 2-1
Sub-rule 2-2-1
Rule 4
1K
DKA
B
C
D
E’
A’
B’
C’
M’
D’
Cracked in
reloading
direction
DM
mM
mθ
M
Sub-rule 2-1
Sub-rule 2-2-2
Rule 4

52. 52
Sub-rule 3-1: The response point follows the unloading stiffness DK ,
( )
'
( )
'
' (1 ')
y A m
A
y A y
y c m
B
y c y
D A B
M M
K
M M
K
K K K
γ
γ
θ
θ θ θ
θ
θ θ θ
λ λ
−
−
−
=
−
−
=
−
= + −
where yielding point Y ( ,y yMθ ) is on the side of
maximum response point M ( ,m mMθ ) and
cracking point C’ ( ' , 'c cMθ ) is on the other side.
Characteristic point D is defined as the
intersection of the unloading line MD and line AB
connecting characteristic points A and B.
Moment DM at characteristic point D is
determined by
m
m B
D
D A
A
A B
D
M
K
M M
M
K
θ θ
θ θ
− −
=
− −
The response is elastic between unloading point M and characteristic point D.
If the response point reaches the unloading point M, the response point follows Rule 3 for
loading on the third skeleton line.
If the response point reaches the
characteristic point D ( ,D DMθ ) of unloading,
the response point follows Sub-rule 3-2.
Sub-rule 3-2: The response point follows
Sub-rule 3-2-1 if no cracking has taken place in
the direction of reloading, Sub-rule 3-2-2 if
cracking has taken place in the direction of
reloading.
Sub-rule 3-2-1: The response point
moves elastically on line DE’ with
unloading stiffness EK , where unloading
stiffness is defined as
1 1
1
' ( )
'
( )
'
' ' (1 ')
m
y
y c m
B
y c y
E B
K K
M M
K
K K K
γ
γ
θ
θ
θ
θ θ θ
λ λ
−
−
=
−
=
−
= + −
Point E’ is defined on the unloading line at
moment level equal to moment 'AM of
characteristic point A’ on the initial
stiffness.
A
C
Y
M
B
'AM
Y’
C’
D
DM
DK
E’
EK
A’
No cracking on
opposite side
Sub-rule 3-1
Sub-rule 3-2-1
Rule 4
C
Y
M
A
D
B
DK
mθyθ
yM
uM
Sub-rule 3-1
Rule 3

53. 53
The previous response point M’ on the side of point E’ is defined as the yielding point Y’.
The unloading stiffness 'DK from the yield point is defined as
' '
' '
'
'
' (1 ')
y A
A
y A
y c
B
y c
D A B
M M
K
M M
K
K K K
θ θ
θ θ
λ λ
−
=
−
−
=
−
= + −
Characteristic point D is defined as the intersection of the unloading line MD and line AB
connecting two characteristic points A and B. The moment DM at point D is given by
y
y B
D
D A
A
A B
D
M
K
M M
M
K
θ θ
θ θ
− −
=
− −
If the response point reaches point D, the response point follows Sub-rule 3-1.
If the response point reaches point E’, the response point follows Rule 4.
Sub-rule 3-2-2: The response
point moves elastically on line DE’
with unloading stiffness EK . The
unloading stiffness EK is defined
by
1 1
1
' ( )
'
( )
'
' ' (1 ')
m
y
y c m
B
y c y
E B
K K
M M
K
K K K
γ
γ
θ
θ
θ
θ θ θ
λ λ
−
−
=
−
=
−
= + −
Point E’ is defined on the
unloading line at moment level
equal to moment level 'DM of
the characteristic point D’. The
characteristic point D’ and its
moment 'DM were defined
during previous unloading from
point M’ on the skeleton curve in
the opposite direction under
Sub-rule 2-1 or 3-1.
If the response point reaches point D, the response point follows Sub-rule 3-1.
If the response point reaches point E’, then the response point follows Rule 4.
Rule 4: The response point moves on line E’M’ toward the previous maximum response point M’
( ' , 'm mMθ ) in the direction of reloading. The characteristic point E’ is defined either in Sub-rule 2-2
or 3-2.
A
B
C
D
Y
M
C’
A’
Y’M’
B’
D’
E’
EK
Cracking in reloading
direction
Sub-rule 3-1
Sub-rule 3-2-2
Rule 4

54. 54
When the response point reaches the previous maximum response point M’ in the direction of
reloading, the response point follows either Rule 2 if no yielding has taken place or Rule 3 if yielding
has taken place in the direction of reloading.
If unloading takes place at point N’ before reaching the previous maximum response point, the
response point follows Sub-rule 4-1.
1EK K=
DKA
B
C
D
E’
A’
B’
C’
M’
D’
DM
F’
G
N’
'DK
'mM
'mθ
A
B
C
D
Y
M
C’
A’
B’
D’
Y’M’
G’
E
N
F
N’
F’
G
Sub-rule 4-1: The response point moves on line N’F’ with unloading stiffness 'DK where
point N’ is an unloading point before the response point reaches the previous maximum
response point M’ in Sub-rule 2-2 or
3-2. Point F’ is defined on the
unloading line N’F’ at moment level
equal to moment 'DM of
characteristic point D’. The
unloading stiffness 'DK ,
characteristic point D’ and its
moment 'DM were defined during
previous unloading from point M’
( ' , 'm mMθ ) on the skeleton curve in
Sub-rule 2-1 or 3-1.
The response point follows line
N’F’ during reloading and
unloading.
If the response point reaches
point N’ during reloading, it follows
Rule 4.
If the response point reached
point F’ during unloading, the
response point follows Sub-rule 4-2.
A
C
Y
M
B
'AM
'DM
B’
Y’
C’
DDM
DK
E’
EK
N’
A’'DK
D’
DK
F’
G