A ModifiedparallelIWANmodelforcyclichardeningbehaviorofsand

8/2/2019 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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Soil Dynamics and Earthquake Engineering

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
http://www.sciencedirect.com/science/journal/sdeehttp://www.elsevier.com/locate/soildynhttp://dx.doi.org/10.1016/j.soildyn.2008.06.008mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.soildyn.2008.06.008http://www.elsevier.com/locate/soildynhttp://www.sciencedirect.com/science/journal/sdee


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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

ARTICLE IN PRESS

k1 k2 k3 k4

1*

2*

3*

4*

Fig. 1. Parallel IWAN model with four elements.

J.-S. Lee et al. / Soil Dynamics and Earthquake Engineering 29 (2009) 630640 631


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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

ARTICLE IN PRESS

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.

J.-S. Lee et al. / Soil Dynamics and Earthquake Engineering 29 (2009) 630640632


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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)

ARTICLE IN PRESS

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)

ARTICLE IN PRESS

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

ARTICLE IN PRESS

-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.

ARTICLE IN PRESS

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,

ARTICLE IN PRESS

-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


-20

-10

0

10

20

Experimental

Hardening Model


-20

-10

0

10

20

Experimental

Without Hardening


-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.

References

[1] Fung YC. Foundations of solid mechanics. Englewood Cliffs, NJ: Prentice-Hall;1965.

[2] Ishihara K. Soil behavior in earthquake geotechnics. New York: OxfordUniversity Press Inc.; 1965.

[3] Kondnor RL, Zelasko JS. A hyperbolic stressstrain formulation of sands. In:Proceedings of the 2nd Pan American conference on soil mechanics andfoundation engineering, 1963. p. 289324.

[4] Duncan JM, Chang CY. Nonlinear analysis of stress and strain in soils. ProcASCE 1970;96(SM5):162953.

[5] Purzin AM, Shiran A. Effects of the constitutive relationship on seismicresponse of soils. Part I. Constitutive modeling of cyclic behavior of soils. SoilDyn Earthquake Eng 20 00;19:30518.

[6] Iwan WD. On a class of models for the yielding behavior of continuous andcomposite systems. J Appl Mech ASME 1967;34:6127.

[7] Van Eekelen HAM, Potts DM. The behavior of Drammen clay under cyclicloading. Geotechnique 1978;28:17396.

[8] Valanis KC, Read HE. A new endochronic plasticity model. In: Pande GN,Zienkiewicz OC, editors. Soil mechnicstransient and cyclic loads, New York,Wiley, 1982. p. 375471.

[9] Bazant ZP, Ansal AM, Krizek RJ. Endochronic models for soils. In: Pande GN,Zienkiewicz OC, editors. Soil mechnicstransient and cyclic loads, New York,Wiley, 1982. p. 41938.

[10] Purzin A, Frydman S, Talesnick M. Normalized non-degrading behavior of softclay under cyclic simple shear loading. J Geotech Eng ASCE 1995;121(12):83643.

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10.10.01 10 10.10.01 10

Period(sec)

0.00

0.10

0.20

0.30

Spectra

lAcc.

(g)

Rock Outcrop

Equivalent Linear

Nonlinear

Nonlinear Hardening

= -0.02, = 0.3

Hachinohe EarthquakeOLE Level

= 5%

Period(sec)

0

0.2

0.4

0.6

0.8

Spectra

lAcc.

(g)

Rock Outcrop

Equivalent Linear

Nonlinear

Nonlinear Hardening

= -0.02, = 0.3

Hachinohe Earthquake

CLE Level

= 5%

Fig. 15. Pseudo-absolute acceleration response spectra of surface motion.

1612840

Time (sec)

-0.2

-0.1

0

0.1

0.2

Acc.

(g)

Hachinohe Earthquake

(PGA is Scaled to 0.15g)

Fig. 14. Acceleration waveform of the Hachinohe Port earthquake.



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[11] Muravskii G, Frydman S. Site response analysis using non-linear hystereticmodel. Soil Dyn Earthquake Eng 1998;17:22738.

[12] Hashiguchi K, Chen Z-P. Elasto-plastic constitutive equation of soils with thesub-loading surface and the rotational hardening. Int J Numer Anal MethodsGeomech 1998;22:197227.

[13] Woodward PK, Molenkamp F. Application of an advanced multi-surfacekinematic constitutive soil model. Int J Numer Anal Methods Geomech1999;23:19952043.

[14] Idriss IM, Dobry R, Singh RD. Nonlinear behavior of soft clays during cyclic

loading. J Geotech Eng Div ASCE 1978;104(GT12):142747.[15] Vucetic M. Normalized behavior of clay under i rregular cyclic loading. CanGeotech 1990;27(1):2946.

[16] Rao SN, Panda AP. Non-linear analysis of undrained cyclic strength of softmarine clay. Ocean Eng 1999;26:24153.

[17] Nelson RB, Dorfmann A. Parallel elasto-plastic models of inelastic materialbehavior. J Eng Mech 1995;121(10):108997.

[18] Finn WDL, Bhatia SK, Pickering DJ. The cyclic simple shear test. In: Pande GN,Zienkiewicz OC, editors. Soil mechnicstransient and cyclic l oads, New York,Wiley, 1982. p. 583607.

[19] Talesnick M, Frydman S. Irrecoverable and overall strains in cyclic shear ofsoft clay. Soils Found 1992;32(3):4760.

[20] Kim DS. Deformation characteristics of soils at small to intermediatestrains from cyclic tests. Ph.D. dissertations, University of Texas, Austin,1991.

[21] Vucetic M. Cyclic threshold shear strains in soils. J Geotech Eng ASCE1994;120(12):220828.

[22] Kim DS, Choo YW. Cyclic threshold shear strains of sands based on pore waterpressure buildup and variation of deformation characteristics. Int J OffshorePolar Eng (IJOPE) 2006;16(1):5764.

[23] Matasovic N, Vucetic M. A pore pressure model for cyclic straining of clay.Soils Found 1992;32(3):15673.

[24] Dobry R, Yokel FY, Ladd RS. Liquefaction potential of over-consolidated sandsin moderately seismic areas. In: Proceedings of conference on earthquakes

and earthquake engineering in Eastern US, vol. 2, Ann Arbor, MI, Knoxville,TN, Ann Arbor Science Publisher Inc., 1981. p. 64364.[25] Finn WDL. Dynamic response analyses of saturated sands. In: Pande GN,

Zienkiewicz OC, editors. Soil mechnicstransient and cyclic loads, New York,Wiley, 1982. p. 10531.

[26] Ni SH. Dynamic properties of sand under true triaxial stress states fromresonant column/torsional shear tests. Ph.D. dissertation, University of Texas,Austin, 1987. p. 421.

[27] Pyke R. Nonlinear soil models for irregular cyclic loadings. J Geotech Eng ASCE1979;105(GT6):71526.

[28] Seed HB, Idriss IM. Soil moduli and damping factors for dynamic responseanalysis. Report no. UCB/EERC-70/10. Earthquake Engineering ResearchCenter, University of California, Berkeley, 1970. p. 48.

[29] M.O.C.T. Korean Seismic Design Standard. Ministry of Construction andTransportation; 1997.

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A ModifiedparallelIWANmodelforcyclichardeningbehaviorofsand

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