A ModifiedparallelIWANmodelforcyclichardeningbehaviorofsand
Transcript of A ModifiedparallelIWANmodelforcyclichardeningbehaviorofsand
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A modified parallel IWAN model for cyclic hardening behavior of sand
Jin-Sun Leea,, Yun-Wook Choo b,1, Dong-Soo Kim b,2
a Climate Change Response Division, National Emergency Management Agency, #1103 Leema Bldg., 146-1 Soosong-dong, Jongno-Gu, Seoul, Republic of Koreab Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, South Korea
a r t i c l e i n f o
Article history:
Received 25 March 2008
Received in revised form
24 June 2008
Accepted 29 June 2008
Keywords:
Bauschingers effect
Cyclic hardening
Cyclic threshold
Irrecoverable strain
Masing rule
IWAN model
Accumulated shear strain
Site response analysis
a b s t r a c t
A modified parallel IWAN model, which includes a cyclic hardening function, is proposed and verified.
The proposed model consists of elasto-perfect plastic and isotropic hardening elements. The model is
able to predict cyclic hardening behavior through the adjustment of the internal slip stresses of its
elements beyond the cyclic threshold, and satisfies Bauschingers effect and the Masing rule with its
own behavior characteristics. The cyclic hardening function is developed based on the irrecoverable
plastic strain (accumulated shear strain) of dry sand during shearing, which is assumed to be a
summation of shear strain beyond the cyclic threshold. Symmetric-limit cyclic loading and irregular
loading tests were performed to determine model parameters and to verify the behavior of the
proposed model. Finally, a one-dimensional site response analysis program (KODSAP) is developed
by using the proposed model. The effects of cyclic hardening behavior on site response are evaluated
using KODSAP.
& 2008 Elsevier Ltd. All rights reserved.
1. Introduction
In order to perform seismic site response and/or soil structure
interaction analyses, the development of reliable constitutive
models that can predict the cyclic stressstrain behavior of soil at
small to intermediate strain ranges is necessary. The most
important behavior characteristics of soil during cyclic loadings
are: (i) a nonlinear stressstrain relationship, (ii) an apparent
reduction in yield stress when loads are reversed, or Bauschingers
effect [1], and (iii) changes in soil properties not only with
shearing strain but also with the progression of cycles, called
cyclic hardening or degradation [2].
A number of approaches have been used to account for the
above phenomena. Mathematical functions and plasticity modelshave been able to predict nonlinear stressstrain behavior,
including Bauschingers effect, under cyclic loading conditions.
Some of the most well-known mathematical function models
are hyperbolic/exponential functions [3,4] and four-parameter
models, which are known as RambergOsgood (RO) models.
Purzin and Shiran [5] suggested a logarithmic function model to
improve the accuracy of the hyperbolic and RO models. In
general, the mathematical function models need a specific
behavioral rule, such as the Masing rule, that defines the
unloading and reloading curves because most mathematical
function models have been developed using backbone curves as
un/reloading curves.
Many plasticity models, such as two surface, kinematic
hardening, and nested yield surface models, have also been
able to predict nonlinear stressstrain behavior, including
Bauschingers effect under cyclic loading, without a specific
behavioral rule. Among these plasticity models, the nested yield
surface model offers great versatility and flexibility for describing
any observed material behavior, although it suffers from inherent
storage inconveniences.
The original IWAN model is based on the assumption that ageneral hysteretic system can be constructed from a large number
of ideal elasto-plastic elements having different yield levels [6].
The IWAN model consists of a collection of perfectly elastic
spring and rigid-plastic or slip elements arranged in either a
seriesparallel or parallelseries combination. The IWAN model
can be categorized as a nested yield surface model when it is
expanded into three-dimensional space.
The above cyclic models have been able to predict material
behavior well, except for soils where the stressstrain relationship
changes with a progressive number of loading cycles. In the case
of soil, changes in volume or rearrangement of particles can occur
during a load repetition beyond a cyclic threshold and can
influence the stressstrain relationship. In order to predict these
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0267-7261/$- see front matter & 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.soildyn.2008.06.008
Corresponding author. Tel.: +822 2100 5494; fax: +82 2 2100 5499.
E-mail addresses: [email protected] (J.-S. Lee), [email protected]
(Y.-W. Choo), [email protected] (D.-S. Kim).1 Tel.: +8242 8695659; fax: +8242 8693610.2 Tel.: +82 42869 3619; fax: +8242 8693610.
Soil Dynamics and Earthquake Engineering 29 (2009) 630640
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soil characteristics, many soil models have been developed since
the 1970s. These may be divided into two types. One type uses
cyclic parameters based on volume changes with shear deforma-
tion, and was developed by Van Eekelen and Potts [7], Valanis
and Read [8], Bazant et al. [9], Purzin et al. [10], Muravskii and
Frydman [11], Hashiguchi and Chen [12], and Woodward and
Molenkamp [13]. The other type uses cyclic parameters as a
number of loading cycles, and was developed by Idriss et al. [14],Vucetic [15], and Rao and Panda [16].
The first researchers to seriously consider the IWAN
model were Purzin et al. [10]. They proposed the modified
serial IWAN model for cyclic degradation of clays, with cyclic
degradation a function of the observed pore water pressure.
The second researchers were Rao and Panda [16], who pro-
posed the modified serial IWAN model for cyclic degradation
of clays with cyclic degradation as a function of the number of
cycles.
The models in which cyclic hardening/degradation varies as a
function of the number of cycles show a problem of whether the
cyclic hardening/degradation function can obey a superposition
rule when the models are subjected to transient loadings (Miner
rule [7]), and show consistent behavior only at one point in eachloading cycle. Therefore, they are unable to account for cyclic
hardening and degradation within a single cycle [10]. For this
reason, it is preferable to use a continuous relationship based on
volume change or shearing deformation of soil as a cyclic
hardening/degradation function. However, another problem with
the models using volume changes or shearing deformation as
their cyclic hardening/degradation function is the difficulty in
determining the function parameters and the irrecoverable strain,
which have generally been used as a concept not of the cyclic
threshold but of the elastic threshold.
For the reasons mentioned above, in this paper the parallel
IWAN model is modified to represent the cyclic hardening
behavior of dry sand by the inclusion of isotropic hardening
elements connected in parallel. The advantages of the parallel
IWAN model are that it is able to predict hysteretic behavior,
including Bauschingers effect, with its own behavior character-
istics without any additional behavior rules, and it obeys
Druckers stability postulate even if a system comes into localized
failure modes [17]. The behavior of the isotropic hardening
elements is controlled by the cyclic hardening function, which
uses accumulated shear strain as a control parameter. Material
parameters of the proposed model can be determined from
symmetric-limit cyclic loading tests (torsional shear tests). The
cyclic hardening control parameter, the accumulated shear strain,
was defined as a summation of shear strain beyond the cyclic
threshold under continuous shearing. The proposed model was
verified with experimental results. A nonlinear site response
analysis program (KODSAP) was developed using the proposed
model. Finally, the effect of cyclic hardening on site responseanalysis was evaluated.
2. The original parallel IWAN model
The original parallel IWAN model, which represents the elasto-
plastic behavior of composite materials, was proposed by Iwan
[6]. Thus, this section is mainly extracted and quoted from the
paper of Iwan [6]. The IWAN model consists of a collection of
perfect elastic and slip elements arranged in either a series
parallel or a parallelseries combination. A four element
parallelseries model is shown in Fig. 1.
Each element consists of a linear spring with stiffness ki in
series with a coulomb or slip damper that has maximumallowable force, tni . The initial loading behavior is described
by Eq. (1):
t Xni1
kig X4
in1
ti (1)
where the summation from 1 to n includes all of those elements
that remain elastic after a loading of deflection g, and thesummation from n+1 to 4 includes all of those elements that
have slipped or yielded. The cyclic behavior of the aforementioned
model is presented in Fig. 2.
In general, both the slip stress tni and the linear elastic stiffnesski in Eq. (1) could be distributed parameters, and the slip stresses
tni are distributed while the elastic stiffness remains constant as k.Let the number of elements become very large so that tni can bedescribed in terms of a distribution function, jtn, wherejtn dtn is the fraction of the total slip elements having a slipstress between tn and tn dtn. Then, Eq. (1) becomes
t kgZ1
kgjtn dtn
Zkg0
tnjtn dtn (2)
If all of the elements have the same elastic stiffness k, and if thetotal number of elements is M, the model parameters can be
evaluated as follows.
Within the first loading step (g 0 to g1), all elements remainelastic. Thus, the elastic stiffness of each element, k, can be
calculated by dividing the initial stiffness t1/g1 by the totalnumber of elements, M, because all elements are assumed to have
the same elastic stiffness:
k t1=g1
M(3)
The number of elements ctni , which remain elastic betweenloading steps i1 and i, can be calculated by dividing the slope Kiby the element stiffness k as follows:
Ki ti ti1gi gi1
; ctni Ki
k(4)
The slip stress distribution function jtn can be determined bycalculating the difference of the number of elements that remain
elastic at each step:
jtni ctn
i ctn
i1 Kik
Ki1
k
ti ti1=gi gi1 ti1 ti=gi1 gi
k(5)
Then, the slip stress of each element can be calculated as follows:
tni kgi (6)
where i 1$N.
If the system is initially loaded to some state of stress andstrain denoted by tm and gm, unloading to a state tm and gm
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k1 k2 k3 k4
1*
2*
3*
4*
Fig. 1. Parallel IWAN model with four elements.
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and reloading to tm will result in a symmetrical hysteresis loop,as shown in Fig. 3.
The stressstrain relation for the unloading loop can be
explained by the combination of three groups of slip elements:
(i) those elements that did not slip upon initial loading and
therefore remain in an unyielded state, (ii) those elements that
yielded in a positive direction upon initial loading but have
stopped slipping, and (iii) those elements that yielded in apositive direction upon initial loading but have now yielded in a
negative direction [6]. Thus, for the parallelseries model, the
stressstrain relation for the lower unloading curve becomes
t
Zkgmg=20
tnjtn dtn
Zkgmkgmg=2
kg kgm tnjtn dtn
kgZ
1
kgm
jtn dtn (7)
The reloading loop curve will have the opposite sign to the
above equation. The integration terms (gmg)/2 in the aboveequation satisfy the Massing rule. Eq. (7) becomes identical to the
corresponding initial loading curve, Eq. (2), when transforming
g0 (gmg)/2 to g0, because the integration term (gmg)/2 meansthat the elastic range of the elements increases by a factor of two
when the element behaves in unloading or reloading. Also, one
can easily notice that this type of model can represent
Bauschingers effect without any behavioral rule because it
satisfies the Masing rule.
3. Development of the modified parallel IWAN model for
cyclic hardening
3.1. Cyclic threshold
When most geological materials are loaded cyclically, the shear
modulus will increase or degrade with increasing loading cycles;
fully saturated sand and clay will show a decrease in shear
modulus (cyclic degradation), while unsaturated sand will show
an increase in shear modulus (cyclic hardening). Previous studies
have shown that such a variation in shear modulus occurs with a
change of volume or pore water pressure of a specimen [18,19].
It is important to note that this variation occurs only if the
amplitude of the cyclic shear strain exceeds a certain thresholdvalue, which is called the cyclic threshold. Generally, two methods
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k1k2 k3
k4
= 1* + 2* + 3* + 4* = 1* + 2* + 3* +(k4)3 = 1* + 2* +(k3+k4)2
= 1* + (k2+k3+k4)1
= -1*+ 2* - k2(5-4)+ 3* - k3(5-4) + 4* - k4(5-4)
= -1
*-
2
*+
3
*- k
3
(5
-3
) + 4
*- k
4
(5
-3
)
= -1* -2* -3* + 4* - k4(5-2) = -1* -2* -3* -4*
1 2 3 4 5
1*2*3*4*
Fig. 2. Behavior of parallel IWAN model with four elements.
m
-m
K
1
mm
Fig. 3. Stressstrain relationship of symmetric-limit cyclic loading.
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have been reported to determine the cyclic threshold of geological
materials. One is by measuring the changes of volume or pore
water pressure in a specimen, and the other is by measuring the
changes of shear modulus according to the loading cycles. At small
strains, the change of shear modulus with the number of loading
cycles can be detected more precisely than the changes of volume
or pore water pressure increment, due to the lack of precision of
measurement devices. Consequently, the cyclic threshold ofvarious soils determined based on the changes of volumetric
strain or pore water pressure increment is slightly larger than that
based on the changes of shear modulus [2022].
Matasovic and Vucetic [23] further developed this idea and
suggested a pore water pressure model for clay, which exhibits
cyclic degradation as a function of cyclic threshold. In this study,
the cyclic threshold gct was determined at a strain wheredeviation from the backbone curve occurs, and was used as a
judgment for controlling the cyclic hardening behavior when a
dry sand undergoes cyclic loadings.
3.2. Accumulated shear strain
In the past several decades, many researchers have developedspecific relationships between the amount of cyclic hardening/
degradation and pore water pressure, volume change, or devia-
toric plastic shear strain [10,24,25]. In classical plasticity theory,
most cyclic soil models have used deviatoric plastic shear strain as
the hardening/degradation control parameter by extraction of the
elastic strain from the total strain. However, in most soils, the
elastic threshold is different than the cyclic threshold, and this
difference has required the measurement of the cyclic threshold
through experiments.
In this study, an irrecoverable shear strain generated under
cyclic loading (accumulated shear strain) is calculated using the
concept of the cyclic threshold. The accumulated shear strain, gacc,is defined as the summation of shear strain beyond the cyclic
threshold, and is used as a cyclic hardening control parameter.Following the Massing rule, strain beyond the cyclic threshold on
the unloading and reloading curve will be 1/2 that of a backbone
curve. The procedure for calculating the accumulated shear strain
is shown schematically in Fig. 4.
The accumulated shear strain during N loading cycles can be
calculated by a summation of the accumulated shear strain
generated in each cycle:
gacc XN
i1
giacc (8)
3.3. Modified IWAN model with isotropic hardening elements
In this paper, a parallel IWAN model with isotropic hardening
elements is proposed to represent the behavior of dry sand under
cyclic loading. The proposed model has two types of elements as
shown in Fig. 5. One type is elasto-perfect plastic elements, which
have smaller slip stress than the threshold slip stress, and the
other is isotropic hardening elements, which have larger initial
slip stress than the threshold slip stress.
Typical behaviors of the two types of elements are shown in
Fig. 6. The threshold slip stress is defined as a slip stress that
corresponds to the cyclic threshold, as indicated in Eq. (9):
tnt ktgct (9)
where kt is the element stiffness of the threshold element and gct isthe cyclic threshold. In order to represent cyclic hardening
behavior, a slip stress distribution function jtn in the originalparallel IWAN model can be modified into a function of the cyclic
hardening parameters such as strain, pore water pressure, and
number of loading cycles. In this study, the accumulated shear
strain is used as a cyclic hardening control parameter; the slip
stress distribution function jfn becomes Eq. (10):
jtngacc jtn0 Hgacc (10)
where tn0 is the slip stress when the soil does not experience
any accumulated shear strain and H(gacc) is a cyclic hardeningfunction, which increases slip stresses of isotropic hardeningelements, corresponding to the accumulated shear strain. Then,
the stressstrain relation of backbone curve becomes Eq. (11):
t kg
Z1kg
jtngacc dtn
Zkg0
tnjtngacc dtn (11)
The above backbone curve shows a unique stressstrain
behavior independent of the number of loading cycles below the
cyclic threshold, whereas above the cyclic threshold, it shows
stiffer behavior (Fig. 7).
The variation of slip stress at the ith element, Dti , can beexpressed as an exponential function with its argument of the
accumulated shear strain (Eq. 12):
tni gacc a bekgacc (12)
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2tc
Shear Strain (%)
ShearStres
s(kPa)
O
B
A
2 1
acc2 = ((2-1)- 2tc)/2
tc0
acc1 = 1-tc
Fig. 4. Calculation of accumulated shear strain.
k2 kM-1 kMkt
Elasto-perfect plasticelements Isotropic hardening elements
k1
1* 3*2* 4*t*
+H(acc) +H(acc)
Fig. 5. Parallel IWAN model with isotropic hardening elements.
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where a, b, and k are material parameters, and gacc is theaccumulated shear strain. As indicated in Eq. (12), the material
parameter k must be positive in order to represent an increase ofslip stress with increasing accumulated shear strain, gacc. Thesummation of a and b becomes the slip stress of an isotropichardening element when a soil does not experience any
accumulated shear strain. Then, the cyclic hardening function
can be derived by calculating the difference between the slip
stress at a given accumulated shear strain and the slip stress at
unhardened state as follows:
Hgacc tngacc t
n0 bekgacc 1 (13)
By substituting Eq. (13) into Eq. (2), the stressstrain relation of
the backbone curve becomes Eq. (14). The same procedure holds
for an unloading and a reloading curve:
t kgZ1
kgjtngacc dt
n
Zkg0
tnjtngacc dtn
kgZ1
kgjtn0 Hgacc dt
n
Zkg0
tnjtn0 Hgacc dtn
kgZ1
kgjtn0 bekgacc 1 dtn
Zk
0tnjtn0 bekgacc 1 dtn (14)
3.4. Effects of model parameters on cyclic hardening behavior
In order to find the physical meaning of the model parameters
b and k, the proposed model was simulated for the twoparameters. First the parameter b was varied from 0.01 to
0.02 while the parameter k was kept at a constant value of0.3. Next, the material parameter k was varied from 0.1 to 0.6while the parameter b was kept at a constant value of 0.01. The
cyclic stressstrain behaviors of the proposed model under
successive cyclic loadings up to a shear strain of 0.3%, with the
variation of the parameters b and k, are plotted in Fig. 8.As shown in Fig. 8, the final amount of cyclic hardening increased
as the absolute value of b increased; however, the convergence
speed to an asymptotic loop was unchanged. The hysteresis loop
converged rapidly to an asymptotic loop as the parameter kincreased; however, the asymptotic loop was identical independent
of the parameter k. It should be noted that the material parameter bcontrols the final amount of cyclic hardening and the material
parameter k controls the convergence speed to the asymptotichysteresis loop with accumulated shear strain.
4. Application of the modified parallel IWAN model
The model parameters b and k can be derived from asymmetric-limit cyclic loading test. The successive un/reloading
curves obtained from the test can be decomposed into backbone
curves by adopting the Massing rule as shown in Fig. 9(a). In order
to obtain a slip stress variation of the isotropic hardening
elements according to the accumulated shear strain, strainreversal points were used as datum points.
For example, if the strain reversal point of the ith un/reloading
curve is gri ; tri as shown in Fig. 9(b), the corresponding shear
stress at the strain reversal point of the backbone curve can be
calculated by Eq. (15):
t kgri
Z1kgr
i
jtn dtn Zkgr
i
0
knjtn dtn (15)
So, the decomposed ith un/reloading curve in Fig. 9(b) can be
derived from the stressstrain relationship of the backbone curves
with increasing slip stresses of the isotropic hardening elements
by the amount ofDtn as follows:
tri kgriZ1
kgri
jtn Dtn dtn Zkgri
0kn
jtn Dtn dtn (16)
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Shear Strain
ElementS
tress f *
- f *
k1
Elements under Threshold Element
Shear Strain
ElementS
tress f *
k1
f *+ H(acc)
Elements over Threshold Element
Fig. 6. Cyclic behavior of two types of elements.
Increase of
Loading Cycles
Hardening
Backbone Curve
(Initial Loading Curve)
t
c(Cyclic Threshold Strain)
Fig. 7. Deformable backbone curve behavior of the modified parallel IWAN model.
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The accumulated shear strain at a datum point can be
calculated as a summation of the accumulated shear strain, gacc,by using Eq. (8). Then, a variation of slip stress can be acquired
from a calculation of Dtn and gacc on each decomposedun/reloading curve.
The parallel IWAN model has a feature that all elements
experience the same value of strain regardless of their slip
stresses. Thus, the accumulated shear strain of the whole system
can be calculated easily by using that of the threshold element. Ifthe threshold element yields plastic strain, a summation of plastic
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-0.4 0.4
Shear Strain (%)
-600
-400
-200
0
200
400
600
ShearStress(kPa)
ShearStress(kPa)
ShearStress(kPa)
= -0.01= 0.3
-600
-400
-200
0
200
400
600
= -0.02= 0.3
-400
-200
0
200
400
ShearStress(kPa)
= -0.01= 0.1
-400
-200
0
200
400
= -0.01
= 0.6
-0.2 0 0.2
-0.4 0.4
Shear Strain (%)
-0.2 0 0.2-0.4 0.4
Shear Strain (%)
-0.2 0 0.2
-0.4 0.4
Shear Strain (%)
-0.2 0 0.2
Fig. 8. Cyclic hardening behavior of the proposed model with varying b and k.
Fig. 9. Determination ofDtn from backbone and decomposed un/reloading curve.
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strain of the threshold element can be considered the accumu-
lated shear strain of the whole system. Typical variations of slip
stress with accumulated shear strain are shown in Fig. 10.
The proposed exponentially increasing function (Eq. (13)) fits
the experimental data well.
5. Model verification
In this paper, a Stokoe type fixed-free torsional shear device
[20,26] was used to perform the cyclic loading tests. In order toverify the proposed cyclic hardening model, two types of tests
were performed. One was a symmetric-limit cyclic loading test
and the other was an irregular loading test. The symmetric cyclic
loading tests were performed to obtain the model parameters,
and to investigate the model behavior. The irregular loading tests
were also performed to verify the applicability of the proposed
model under transient loading conditions. Two types of dry sands
(Kum-Kang and Toyoura sands) were tested. The specimens were
formed by air pluviation and confining pressures were applied by
a vacuum device. A brief description of the engineering properties
of the tested sands is listed in Table 1.
5.1. Verification with symmetric-limit cyclic loading tests
The applicability of the proposed model under symmetric-
limit cyclic loading conditions was investigated. Hardening
parameters of the proposed model, b and k, were determinedfrom symmetric-limit cyclic loading tests. The tests were
performed by applying three successive loadings of 30 cycles
using a sinusoidal waveform with a loading frequency of 0.06 Hz.
The number of elements for the proposed model was identical to
the number of data samples acquired from the initial experi-
mental loading curve, and ranged from about 40 to 45 for each
test. Representative experimental results are compared with
behaviors of the proposed model in Fig. 11.
The experiments show that the proposed model estimates
the cyclic hardening behavior of the tested sands very well. At the
first cycle, all comparisons show excellent agreements. Thesecomparisons show clearly that the proposed model is able to
represent cyclic hardening behavior very well within a loop
(initial loading and subsequent un/reloading) as well as with the
number of loading cycles.
5.2. Verification with irregular loading test
Irregular loading tests were performed to verify the appli-
cability of the proposed model under transient loading conditions
such as an earthquake. The shape of the irregular loading was as
shown in Fig. 12, which was suggested by Pyke [27]. The
maximum amplitude of the driving forces on the soil specimenwas fixed at 14 kN, and the tests were performed with five
successive irregular loading shapes.
The cyclic hardening parameters (b and k) and an initialloading curve were derived from symmetric-limit cyclic
loading tests that had similar testing conditions (relative density
and confining pressure) as the irregular loading test. A compa-
rison of the test conditions between the symmetric-limit
cyclic loading and the irregular loading tests is listed in
Table 2.
The number of elasto-plastic and isotropic hardening elements
for the proposed model was 40. The cyclic behavior of the
proposed model was compared with the irregular loading test
results. In order to determine the effect of the cyclic hardening
function, the model behaviors both with and without the cyclichardening function are included in Fig. 13.
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0
Accumulated Strain (%)
0.04
0.08
0.12
0.16
SlipStress(kPa
)
Experimental
Slip Stress Variation Function
Toyoura Sand, Dr = 54.3%, P0 = 50kPa
f*(acc) = 0.139-0.086e-1.363acc
1 2 3 4 5
Fig. 10. Typical variation of slip stress with accumulated shear strain.
Table 1
Summary of engineering properties of tested sand
Properties Kum-Kang sand Toyoura sand
Classification (USCS) SP SP
Maximum void ratio 0.973* 0.982
Minimum void ratio 0.698* 0.617
Gs 2.65
Cc (coefficient of gradation ) 0.96 1.00
Cu (uniformity coefficient) 2.46 1.29
PI NP NP
D50 (mm) 0.424 0.199
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The cyclic behavior of the model without the cyclic hardening
function can be regarded as that of existing soil models, such as
the original IWAN, hyperbolic, RambergOsgood, and plasticity
models. These comparisons show that the model behaviors
predicted by the proposed model show reasonably good agree-
ment with the experimental results, particularly in the 1st loading
stages. There were some discrepancies between the model
behaviors and the experimental results, and they became largerwith an increase in the loading stage. The possible reasons for the
discrepancies can be explained as follows: (i) the relative densities
of the symmetric-limit cyclic loading test, which were used to
determine the hardening parameters, were slightly different than
those of the irregular loading test, or (ii) the coupling problems
between a top cap plate and a specimen may have generated
biased shifts in the hysteresis loop, which are shown in the lower
part of hysteresis loops in Fig. 13.
6. Application to one-dimensional site response analysis
6.1. Development of one-dimensional site response program
In this study, a one-dimensional nonlinear site response
analysis program KODSAP (Kaist One-Dimensional Site Amplifica-
tion Program) was developed using the proposed model as its
constitutive equation. The developed program uses the direct
numerical integration method and numerical integration is
conducted using Newmarks b method. The bedrock half space
of the soil deposit is treated as a linear elastic or rigid body. The
soil deposit is assumed to be shaken by horizontal shear waves
propagating vertically. The layered soil deposit is converted to a
lumped mass system by lumping one-half of the mass of each
layer at the layer boundaries. The masses are connected by
nonlinear springs with stressstrain properties given by theproposed model for initial loading, subsequent unloading,
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-0.2
Shear Strain (%)
-20
-10
0
10
20
ShearStress(kPa)
ShearStress(kPa)
Experimental
Estimated
1st
Cycle
Kum-Kang Sand, Dr = 54.7%, P0 = 50kPa
Shear Strain (%)
-20
-10
0
10
20
Experimental
Estimated
Kum-Kang Sand, Dr = 54.7%, P0 = 50kPa
-0.08
Shear Strain (%)
-20
-10
0
10
20
ShearStress(kPa)
Experimental
Estimated
Toyoura Sand, Dr = 54.3%, P0 = 50kPa
1st
Cycle
-0.08
Shear Strain (%)
-20
-10
0
10
20
ShearStress(kPa)
Experimental
Estimated
Toyoura Sand, Dr = 54.3%, P0 = 50 kPa
-0.1 0 0.1 0.2-0.2 -0.1 0 0.1 0.2
-0.04 0 0.04 0.08 0.12 -0.04 0 0.04 0.08 0.12
Fig. 11. Comparisons between model behavior and experimental results.
0
Time (sec)
-20
-10
0
10
20
ShearStress(kPa)
20 40 60 80 100
Fig. 12. Irregular loading shape (suggested by Pyke [27])
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and reloading. The proposed model reflects the nonlinear, strain-
dependent, hysteretic, and cyclic hardening and degra-
dation behavior of soil. In addition to the inherent hysteretic
damping, viscous damping can be added independently ifdesired, and they are included here for completeness of the
formulation.
6.2. Effects of cyclic hardening on site response
Parametric studies on site response analysis by employing the
proposed cyclic hardening model (KODSAP analysis) were per-
formed and the results were compared with those obtained from
the nonlinear analysis, and equivalent linear analysis using
program SHAKE91. In the parametric study, the waveform in the
base layer obtained by deconvolution from the recorded wave-
form at the ground surface observed at Hachinohe Port, Japan,
during the 1968 Tokachi-Oki earthquake (Fig. 14), was used as thecontrol motion.
The analysis model layer consisted of a half space with about
40 m of dry sand above bedrock. The shear wave velocity of the
soil profile was assumed to increase linearly from 150 to 340 m/s
with depth in the dry sand layer, considering the effect ofconfining pressure. The shear wave velocity of rock was assumed
to be 760 m/s. The backbone curve suggested by Seed and Idriss
[28] was used to define the backbone curve of dry sand. The
hardening parameter, b, was 0.02 and k was 0.3. A total of 100elasto-perfect plastic and isotropic hardening elements were used
in this parametric study. Rock outcrop accelerations of two types
of waveforms were scaled to a value of 0.06 g (operation level
earthquake) and 0.15 g (collapse level earthquake) based on the
Korean Seismic Design Standard [29].
Pseudo-absolute acceleration response spectra of the ground
surface obtained by equivalent linear analysis, nonlinear analysis,
and nonlinear analysis using the hardening model are shown in
Fig. 15.
In the case of nonlinear analysis using the hardening model,the response spectra tend to increase in the short period range,
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Table 2
Comparison of the test conditions
Type of sand Type of test Relative density (%) Confini ng pressure (kPa) Maximum driving stress (kPa) Sampling rate (s)
Toyoura sand Symmetric-limit loading 54.3 50 14 0.1
Irregular 53.3 50 14 0.01
-0.08
Shear Strain (%)
-20
-10
0
10
20
ShearStress(kPa)
ShearStress(kPa)
ShearStress(kPa)
She
arStress(kPa)
ExperimentalHardening Model
1st
Stage
Toyoura Sand, Dr = 55.7, P0 = 50kPa
-20
-10
0
10
20
ExperimentalWithout Hardening
1st
Stage
Toyoura Sand, Dr = 55.7, P0 = 50kPa
-20
-10
0
10
20
Experimental
Hardening Model
Toyoura Sand, Dr = 55.7, P0 = 50kPa
-20
-10
0
10
20
Experimental
Without Hardening
Toyoura Sand, Dr = 55.7, P0 = 50kPa
-0.04 0 0.04 0.08
-0.08
Shear Strain (%)
-0.04 0 0.04 0.08 -0.08
Shear Strain (%)
-0.04 0 0.04 0.08
-0.08
Shear Strain (%)
-0.04 0 0.04 0.08
5th
Stage5th
Stage
Fig. 13. Comparisons between model behavior and experimental results of irregular loading tests.
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but the changes at long periods were negligible when they are
compared with the nonlinear analysis results. The increase of the
response spectra in the short period range is larger in the collapse
level earthquake than in operation level earthquake. The increase
of response spectra in the short period range is mainly due to the
movement of the natural period of the model layer toward short
periods during the earthquake. The cyclic hardening phenomena
in dry sand cause the natural period of model layer to be shorter.
Thus, it can be concluded that the cyclic hardening behavior of dry
sands can cause an increase of the response spectra in the short
period range, and the increase becomes larger with increasing
peak accelerations.
7. Conclusions
A key aspect of this paper is the development of a cyclicsoil behavior model that can represent three important
dynamic behavior characteristics of dry sand. These are: (i)
nonlinear stressstain relationship, (ii) Bauschingers effect,
and (iii) cyclic hardening. Among the many types of nonlinear
cyclic soil behavior models, the parallel IWAN model is one
that represents both the nonlinear stressstrain relationship
and Bauschingers effect well without any additional behavior
rules.
For these reasons, the modified version of the original parallel
IWAN model is proposed in this paper to represent the cyclic
behavior of dry sand. The original parallel IWAN model was
modified to be able to represent cyclic hardening behavior with
the help of isotropic hardening elements.
The concept of a cyclic threshold strain and an accumulatedshear strain was employed as a cyclic hardening control
parameter. The proposed model showed good agreement with
experimental results, including both symmetric-limit cyclic
loading and irregular loading test results. The effects of cyclic
hardening on earthquake site response was evaluated by a one-
dimensional nonlinear sites response analysis program (KODSAP)
using the proposed model as its constitutive equation. The
analysis results show that the cyclic hardening behavior of dry
sands can cause an increase of the response spectra in the short
period range.
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Period(sec)
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