nonlinear seismic response and damage of reinforced concrete

Journal of Advanced Concrete Technology Vol. 7, No. 3, 439-454, October 2009 / Copyright © 2009 Japan Concrete Institute 439

Scientific paper

Nonlinear Seismic Response and Damage of Reinforced Concrete Ducts in Liquefiable Soils Mohammad Reza Okhovat1, Feng Shang2 and Koichi Maekawa3

Received 19 May 2009, accepted 31 July 2009

Abstract Highly inelastic nonlinear interaction of liquefied soil and underground RC ducts is computationally investigated in view of the structural damage. The required ductility is expected to be insensitive to the risk of liquefaction for normally deposited layers of soil, and the lesser ductility is acceptable for the case of highly liquefiable foundation similar to the seismic isolation. Although the structural nonlinearity has a fewer effect on the uplift of the underground ducts, the amount of main reinforcement may control the structural damage with the same efficiency for both drained and undrained soil deposits.

1. Introduction

Underground facilities as indispensable parts of urban infrastructures are used for various applications ranging from small pipelines to large RC ducts and tunnels. In the past decades, less seismic damage to large under-ground structures was reported compared to on-ground ones although damages to small pipelines were ob-served as early as 1964 in Niigata and Alaska earth-quakes (Hall and O’Rourke 1991). These facts made structural engineers deem that underground RC struc-tures might be rather safe until Hyogoken-Nanbu earth-quake in 1995. This earthquake caused severe damages in some RC underground structures which failed in the mode of shear accompanying deteriorated axial load carrying mechanism (An et al. 1997). Afterward, this catastrophic damage was observed again in the follow-ing earthquakes of 1999 Chi-Chi, Taiwan (Wang et al. 2001) and 1999 Duzce, Turkey earthquakes (O’Rourke et al. 2001). Thus, earthquake-induced damages to un-derground structures and the corresponding earthquake resistant designs have received remarkable attention.

Although the seismic performance of underground structures has been extensively investigated (for exam-ple An et al. 1997; Huo et al. 2005; Hashash et al. 2001), there have been limited studies regarding the liquefac-tion-related seismic performance. Kimura et al. (1995) conducted centrifuge tests to examine the effect of vari-ous countermeasures against liquefaction with a mock-up of underground structures. Koseki et al. (1997a and

1997b) investigated dynamic behaviors of several types of underground structures in liquefiable sand subjected to earthquake excitations. Tamari and Towhata (2003) performed a series of shaking table tests on an alumi-num fixed base structure model embedded in liquefiable soil to propose a solution for determining dynamic soil pressures on a flexible underground structure during liquefaction. In most of these researches, main focus has been addressed to nonlinear behaviors of soil foundation where a simple linear response of built-in structures was assumed. In contrast, this paper aims at deeper views towards the inelastic performances of underground RC structures subjected to seismic excitations in considera-tion of extremely high nonlinearity of liquefied soil foundations.

Seismic actions for practical designs of underground ducts are currently characterized by the forced dis-placement and/or mean strains imposed to the target structure through fictitious springs of soil foundation. First, free-field ground deformations under a specified design seismic event(s) are estimated and second, the underground RC is designed to accommodate these re-quired deformations. This approach is satisfactory if the underground facility is located in a stiffer soil matrix (Hashash et al. 2001).

In this paper, inelasticity and damage of RC ducts like subway tunnels, having interaction with liquefiable soils of extremely high nonlinearity, are targeted in use of the nonlinear dynamic finite element analysis. For investi-gating how much the structural nonlinearity would in-fluence on the entire response, both linear and nonlinear analyses are used to numerically reproduce the under-ground structures. The mechanical properties and di-mensions of structural models are systematically changed to examine their effects on the entire structural performances under ground. All analyses are conducted in both drained and undrained states of pore water. Liq-uefaction may greatly increase in the deformation of soil around structures, but at the same time, the stiffness and

1Ph.D. candidate, Department of Civil Engineering, University of Tokyo, Tokyo, Japan. E-mail:[email protected] 2Ph.D. candidate, State Key Laboratory of Hydro Science and Engineering, Tsinghua University, Beijing, P. R. China. 3Professor, Department of Civil Engineering, University of Tokyo, Tokyo, Japan.

440 M. R. Okhovat, F. Shang and K. Maekawa / Journal of Advanced Concrete Technology Vol. 7, No. 3, 439-454, 2009

internal stress of soil are dramatically reduced as well. A question is raised, what is the resultant of the both kinematics in RC damages?

2. Nonlinear constitutive models

2.1 Constitutive model for reinforced concrete A reinforced concrete material model has been con-structed by combining constitutive laws for cracked concrete and those for reinforcement. The fixed multi-directional smeared crack constitutive equations (Maekawa et al. 2003) are used as summarized in Fig. 1. Crack spacing and diameters of reinforcing bars are implicitly taken into account in smeared and joint inter-face elements no matter how large they are.

The constitutive equations of structural concrete sat-isfy uniqueness for compression, tension and shear transfer along crack planes. The bond between concrete and reinforcing bars is taken into account in the form of tension stiffening model, and the space-averaged stress-strain relation of reinforcement is assumed to represent the localized plasticity of steel around concrete cracks. The hysteresis rule of reinforcement is formulated based upon the Kato’s model (1979) for a bare bar under re-versed cyclic loads. This RC in-plane constitutive mod-eling has been verified by member-based and structural-oriented experiments. Herein, the authors skip the de-

tails of the RC modeling by referring to Maekawa et al. (2003).

2.2 Constitutive model for soil A nonlinear path-dependent constitutive model of soil is essential to simulate the entire RC-soil system. Here, the multi-yield surface plasticity concept (Towhata and Ishihara 1985; Towhata 2008; Maekawa et al. 2003) is applied to formulate the shear stress - shear strain rela-tion of the soil following Masing’s rule (1926).

The basic idea of this integral scheme is actualized to sum up component stresses which may represent micro-scopic events. First, the total stress applied on soil parti-cle assembly, denoted by σij, can be decomposed into the deviatoric shear stresses (sij) and the mean confining stress (p) as,

ijijij ps δσ += (1)

where δij is Kronecker’s delta symbol. Soil is idealized as an assembly of finite numbers of

elasto-perfectly plastic components, which are concep-tually connected in parallel. As each component is given different strengths, all components subsequently begin to yield at different total shear strains, which results in a gradual increase of entire nonlinearity. The nonlinear behavior appears naturally as a combined response of all

Fig. 1 Composition of constitutive models of reinforced concrete (Maekawa et al. 2003).

M. R. Okhovat, F. Shang and K. Maekawa / Journal of Advanced Concrete Technology Vol. 7, No. 3, 439-454, 2009 441

components. Hence, the authors propose the total shear stress carried by soil particles being expressed with re-gard to an integral of each component stress as,

mkl

mkl

mkl

mkl

m

mijm

pij

mpijij

mo

meij

mo

mij

mmo

n

m

mpklkl

mijij

Fds

Fdes

df

dfF

sde

dedeGdeGds

FGss

ε

εε

==

=

−==

= ∑=

,2

)(22

),,,(1

(2)

where εkl and εpkl are total and plastic strain tensors of (k,l) component, Go

m and Fm are the initial shear stiff-ness and the yield strength of the m-th component, (eij, em

pij, emeij) are deviatoric tensors of total strain, those of

plastic and elastic strains of the m-th component, re-spectively. These component parameters can be uniquely decided from the shear stress strain relation of soil under the referential constant confinement (Maki et al. 2005).

In general, the volumetric components may fluctuate and affect the shear strength of the soil skeleton. In real-ity, the shear strength of soil may decay when increasing pore water pressure leads to reduced confining stress of soil particle skeleton. The multi-yield surface plastic envelope may inflate or contract according to the con-finement stress as shown in Fig. 2. It can be formulated by summing up the linear relation of the shear strength and the confinement stress as,

3)(

)tan(

3211

1

σσσ

φχ

χ

′+′+′=′

′−=

=

I

SIcFF

u

mini

m

(3)

where Su is the specific shear strength corresponding to a certain confinement (98kN), Fm

ini is the specified yield strength of the m-th component corresponding to Su, χ is the confinement index, (c,φ) are the cohesive stress and the frictional angel, respectively.

For simulation of the pore water pressure and related softening of soil stiffness in shear, the volumetric nonlinearity of soil skeleton has to be taken into account. The authors simply divide the dilatancy into two com-ponents according to the microscopic events. One is the consolidation (negative dilation) as unrecoverable plas-ticity denoted by εvc. The other is the positive dilatancy associated with alternate shear stress due to the overrid-ing of soil particles, which is denoted by εvd as,

0 03 ( ) ,v v vc vdp K ε ε ε ε ε= − = + (4)

where K0 is the initial volumetric bulk stiffness of soil particles assembly and can be calculated by assuming the initial elastic Poisson’s ratio denoted by ν (=0.2) as,

00 )21(3)1(2 GK

νν

−+

= (5)

The volume reduction of pores among soil particles will cause increasing pore pressure under undrained states, which may lead to liquefaction. According to experiments of sandy soils, the following formulae are adopted as,

( ){ } inivcinippvvc JJ ,,22lim, )(2exp1 εεε −+−−= (6)

klklppp dsdJdJJ ε⋅≡= ∫ 21, 222

(7)

which is represented by the accumulated shear strain invariant of the soil skeleton denoted by J2p (Maekawa and An 2000; Maki et al. 2005), and εv,lim is the intrinsic volumetric compacting strain corresponding to the minimum void ratio as,

{ }( )0.1log1.0

)2exp(16.0

110,lim

,2,lim,

+′=

−−=

I

J

v

inipvinivc

ε

εε (8)

If the relative density of soil is assumed to be Dr, the following relation can be used to inversely decide J2p,ini, which is a constant corresponding to the initial com-pactness of soil particles as,

{ })2exp(1(%) ,2lim,

,inip

v

inivcr JD −−==

εε (9)

The shear provoked dilation, which is path-independent and defined by the updated shear strain intensity J2s, is empirically formulated as,

( )( )

( )lim,

,

2

22

22

015.0

0.25,21

1

v

inivvc

ijijs

s

svd

aeeJ

JaJa

εεε

η

ηε

+=

==

+=

(10)

Fig. 2 Confinement dependent soil model under drained cyclic shear loadings.


Within this scheme, the liquefaction induced nonlin-earity and cyclic dilatancy evolution can be consistently computed. Figure 3 shows the pure shear stress-strain relation and the corresponding pore pressure of perfect undrained soil. Here, the soil characterisitics are repre-sented solely by the initial stiffness and the relative den-sity. The pore water media is assumed to be perfect elas-tic body with no shear stiffness and Biot’s two-phase theory (1962) was coupled with the effective stress model of soil skeleton as stated above. Mean shear stiff-ness decay and the following cyclic mobility are repro-duced as shown in Fig. 3.

The general 3D constitutive model of soil is directly applied to the 2D plane strain field to consider the high confinement along the tunnel axis. Here, out-of-plane strain is computationally forced to zero and the corre-sponding out-of-plane confining stress is computed as a variable together with the in-plane stresses, since the stress states are fully three-dimensional.

The multi-yield surface plasticity model has two main advantages. First, it can easily simulate the shear cyclic responses by means of the simplified algorithm with rather few material constants. Second, the path-dependency of soil can be represented only by the plas-tic strains (em

pij) of all constituent components. As the multi-component scheme has the great similarity to the contact density model of crack shear transfer (Li et al. 1989) and the multi-directional crack modeling of rein-forced concrete (Maekawa et al. 2003), higher stability of computing soil-RC structural interaction is made pos-sible.

The overall experimental verification of the interac-tion analysis was reported (JSCE 2002) in the case of drained soil with RC ducts. The model has been used to simulate the static and seismic behaviors of nonlinear soil-structure systems (An et al. 1997; Nam et al. 2006). Regarding the liquefiable soil-RC interaction, the appli-cability of the models used in this paper was examined and verified by Maki et al. (2005).

2.3 Constitutive model for joint interface It is obvious that the interface property has an influence on the entire soil-structure responses as well. As soil and structure have different nonlinearity, complete contact at the interfaces does not hold. The initial interface of the

soil and the structure is naturally in mostly complete contact under static loads (self-weight and service loads) with lateral earth pressures that soil exerts on the structure. Under reversed cyclic shear due to the seismic excitations, however, rocking of a structure may cause gap-opening and shear sliding along the soil-structure contact especially when cohesive soil is assumed (Maekawa et al. 2003).

In this paper, the contact interface model assuming a bilinear relation for the opening/closure mode is em-ployed. The normal stress of the joint is assumed to be zero in the case of separation. The contact stiffness in closure mode is assigned to be so large as to numeri-cally ensure no overlapping as shown in Fig. 4(a). For the shear sliding mode, the shear stress-slip relation is assumed to be linear till the frictional limit as shown in Fig. 4(b). A couple of contact planes are allowed to slide if the magnitude of the applied shear stress ex-ceeds the frictional limit, which follows the Coulomb’s law (Maekawa et al. 2008). The initial state of the soil-structure interface must be simulated to represent the static earth pressure. This is achieved by performing an analysis that considers the body force of the soil mass alone before applying dynamic actions.

The current joint interface model can be applied to the states of soil liquefaction as well as the drained static and dynamic conditions and pre-liquefied soils. However, due to the dramatically reduced shear stiffness of liquefied soil similar to liquid, a quasi-hydrostatic pressure consequently develops inside the soil founda-tion after liquefaction and it allows less separation to occur between soil and RC joint interfaces. Then, the joint interface modeling occupies minor roles on the structural damage after liquefaction.

3. Simulation of liquefiable soil and RC ducts

3.1 Model properties To investigate the seismic damages of underground RC ducts, a typical subway tunnel section is modeled whose wall and slab dimensions are shown in Fig. 5(a). The center column to mainly support the dead weight of soil overlay has a rectangular cross section of 0.60 × 0.80 m and is idealized as firmly fixed to the upper and bottom

Fig. 3 Confinement dependent soil model under undrained cyclic shear loading.


slabs. The clear distance between two adjacent columns is 3.0 m. The tunnel which is regarded as the standard case is stiffened with 45° haunches at the corners and has a longitudinal reinforcement ratio of 1.1% for side walls and slabs, 1.6% for the column, and web rein-forcement ratio of 0.2% for all elements. The soil de-posit is assumed to be loose sand with a thickness of 15 m which is put on a 5-meter-thick layer of the non-liquefiable soil which lies on the bedrock as shown in Fig. 5(a). The details of material property for rein-forcement, concrete, joint interface and non-liquefiable clayey layer are shown in Table 1.

By assuming the plane strain condition, the finite element mesh used in the analysis is composed of eight-node isoparametric 2D elements. The RC-soil interfacial elements are placed in between the soil and the RC ele-ments. Since the angle of internal friction of the model sand is 30°, the friction angle of the interface is obtained by using the formulae tan-1[(2/3) tanφ], which is about 21°. Totally, 7303 nodes and 2352 finite elements are arranged in the dynamic model. The north-south com-ponent of the rock base acceleration measured at 1995 Kobe earthquake, which is scale-adjusted to 0.3g based on the measurement at Kobe meteorological observatory, is used as the input bed rock motion. It shows the high horizontal ground acceleration with a short period as shown in Fig. 6.

3.2 Boundary conditions The boundary between the soil deposit and the bedrock (see Table 1) is simply assumed to be fixed and would

act as the bottom boundary of the analysis domain at which the earthquake motion is imposed. The ground surface is assumed to be flat and free of loads. The un-derground water level is assumed to be at the level of ground surface so that the entire soil is saturated.

The soil deposit at the far fields should be assumed as the boundary of free shaking. In the shaking table tests of soil-structures, a laminar shear box may be used to simulate the quasi-far-field boundary (Towhata 2008). Here, quasi-far-field elements with a length of 10 m are placed at each extreme side of the analysis domain as illustrated in Fig. 5(b). Both stiffness and unit weight of far-field elements are increased 100 times fictitiously in order to computationally realize the free-excitation which is not substantially affected by the motion of the target domain of high nonlinear interaction with the structures. It should be noted that these fictitious pa-rameters of these two boundary columns result in the same free natural frequencies and dynamic properties as those of the pure soil layers without underground struc-tures. As the far-field mode of seismic motion is lami-nated simple shear, the horizontal length of these boundaries is selected approximately half of the domain height so that the bending deformational mode would not appear. In addition, confinement independent soil elements are used in the quasi-far-field zone in order to prevent the edge collapse in analysis. Consequently, this boundary simply makes the horizontal displacements at both right and left sides equal to each other, which is similar to the case of laminar shear boxes widely used in the shaking table tests.

ωJoint open

closed

Normal stress: σ

Shear stress: τ

δ

Joint closed Joint opened

re-contacted

Ope

n an

d cl

osed

aga

in

−μσ

μσ

(a) Normal response (b) Shear response

Fig. 4 Normal and shear response of linear elastic interface model. (Maekawa et al. 2008).

Table 1 Material properties used.

Non-liquefiable clayey layer* Reinforced Concrete Interface** Initial stiffness 105 MPa Concrete strength 24 MPa Normal modulus 108 kPa/mPoisson ratio 0.20 Unit weight 24 kN/m3 Shear modulus 103 kPa/mDry unit weight 16 kN/m3 Poisson ratio 0.18 Friction angle 21º Friction angle 40 º Steel Young modulus 2.0 x105 MPa Cohesion 0 Cohesion 100 kPa Steel yield stress 240 MPa Relative density 75 %

*) These properties are used for hard base bed of foundation for all parametric studies to numerically avoid liquefaction.**) Simple frictional model used.


Several trial analyses were carried out to determine the size of the analysis domain. Finally, a relatively large analyzed domain (200 m) was specified so as to make the wave reflection negligible.

3.3 Analysis procedure As the seismic analysis of the soil-structure system re-quires an initial stress field of the static equilibrium be-fore dynamic earthquake loads (Liu and Song 2005; Maekawa et al. 2003), the static analysis of drained foundation was firstly performed. In general, repro-duced static soil stress fields might depend on the proc-ess of construction works. But, it was confirmed by the preliminary analyses that the initial earth pressure on the RC ducts has minor influence on the damage of struc-tures when high nonlinearity is induced to both soil and structure under large ground motions.

For investigating the effect of soil liquefaction on the structural damages, several models with and without the

ducts are analyzed in both drained and undrained states of pore water with initial shear stiffness (G0) varying from 12 MPa to 182 MPa as shown in Table 2. The structure is assumed to be located inside the soil at a depth of 4 m without any change in the mesh property of the remaining soil elements. Then, the standard case of the RC duct is analyzed by using linear elastic ele-ments with a Young modulus of 30,000 MPa, Poisson’s ratio of 0.20 and the unit weight of 24 kN/m3, keeping the sandy soil property constant (G0= 35 MPa, Dr=32 % and dry unit weight=14 kN/m3). Finally, a parametric study is performed to investigate the effect of RC nonlinear properties. Wholly, 34 different cases are studied with different reinforcement ratios, different surrounding soil stiffness and two different yield strengths of steel. All these cases are analyzed in both drained and saturated soil conditions.

It should be pointed out that the authors intentionally implement fully undrained conditions for saturated soil

(a) Soil-structure layout

(b) Finite element discretization

Fig. 5 Entire soil-structure system properties.

Fig. 6 Input earthquake motion and its response spectrum.

x

y


elements which may bring about earlier liquefaction than reality, because the aim of this study is to investi-gate the magnitude of damage to RC ducts by liquefi-able soils. As a matter of fact, this assumption is not far from the facts that the required time for drainage of a several-meter-thick sand layer is 10-30 minutes, which is much longer than the duration time of earthquake loading (Towhata 2008).

4. Nonlinear interaction and simulation

4.1 Liquefaction induced damage to under-ground RC ducts

Figure 7 shows the shear response versus the Standard Penetration Test N-value (denoted by SPT) of soil layer for both cases of plain soil and with RC ducts in drained and undrained conditions. The values of maximum shear deformation are also shown in Table 3 for further discussion. Here, the shear deformation of the duct is defined as the drift angle between the upper and bottom slabs. It can be observed that current design approach based on the free-field ground displacement can predict the deformational demand on the underground RC ducts which are embedded in somehow stiffer soil mediums. In soft layers of soil which consists of loosely deposited sand, the required ductility of the duct in structural de-signs becomes smaller than that of the single soil me-dium. This is attributed to the reduced stiffness of the soil foundation compared to that of the RC duct (Fig. 7(a)). More or less, the required ductility of the struc-ture rises according to the reduction of soil stiffness provided that no liquefaction takes place.

Besides, liquefaction may significantly bring about increased soil deformation which indicates large soil strains associated with greatly reduced shear stiffness. Accordingly, the ductility demand on the RC duct under liquefaction becomes larger than those of the drained

Table 2 Material property for liquefiable sandy layer.

NSPT G0* (MPa) K0 (MPa) Dr** (%)1 12 16 25 3 29 39 28 5 44 59 32

10 76 101 40 15 105 140 48 20 132 176 56 25 158 211 65 30 182 243 75

*) G0 is defined according to the empirical formulae as G0 = 12(N-value) 0.8. **) The value of Dr represents how tightly the soil is deposited, and it gives the risk of liquefaction.

Principal tensile strain state of tunnel at maximum shear deformation

(a) Drained condition

Principal tensile strain state of tunnel at maximum shear deformation

(b) Undrained condition Fig. 7 Effect of soil stiffness on the maximum response of averaged shear deformation.

Elastic Concrete cracked

Rebar yielded

Elastic Concrete cracked

Rebar yielded


case if N-value is more than 5. It means that the magni-fied soil deformation by liquefaction and the associated reduced stiffness of the foundation may not be balanced but the former factor gets extensive in view of the RC structural damage. The current designs tend to require further enhanced ductility in liquefiable foundation, which causes denser arrangement of web reinforcement and tends to accompany some difficulty of construction. Anyhow, this trend is consistent with the analytical re-sults as discussed above.

On the contrary, if the soil is much loosely deposited (N-value less than 5), we have dramatically decreased deformations compared to the case of drained founda-tion as shown in Fig. 7(b). The practical point of impor-tance is that the maximum shear deformation finally declines according to the greatly reduced stiffness of the

liquefied soil. This trend is hardly taken into account in current simplified design procedures with linear soil-spring even though its stiffness is made degenerated.

The pore pressure ratio (the excessive pore pressure normalized by the initial effective overburden pressure) above and below the duct are shown in Fig. 8. It can be seen that the liquefaction below the duct is compara-tively minor. This is consistent with the site observation of the past earthquakes. As a matter of fact, the exces-sive uplift flotation of the underground lightweight structures would cause larger shear deformation of sur-rounding soil, which may occasionally cause the de-crease in pore pressure. Under larger shear deformation, normal sand tends to dilate, which shall lead to the low-ering of pore pressure (Liu and Song 2005), that is to say, the cyclic mobility.

Figure 9 shows the comparison of acceleration re-sponses at the ground surface. It can be observed that the acceleration in liquefiable soil is much smaller than that of the drained dry soil. The earthquake energy would be damped out by dramatically reduced shear stiffness of soil skeleton and accompanying larger de-formation which occur at the time of liquefaction. The acceleration response is not affected by the presence of the structure.

4.2 Effect of RC nonlinearity The horizontal and vertical dynamic displacements of the linear elastic duct and the highly nonlinear RC are compared as shown in Fig. 10 and Fig. 11, respectively. The duct exhibits some settlement during the ground motions in the drained condition, while liquefied soil pushes the underground duct upward significantly as observed in the past earthquakes and some laboratory tests (Towhata 2008). Then, some countermeasures have been discussed to reduce the uplift of underground structures in liquefiable soils.

Table 3 Maximum responses of averaged shear deformation.

Maximum Shear Deformation

Drained Condition Undrained ConditionSPT N-

value Free Field

with Tun-nel

Free Field

with Tun-nel

1 1.20 0.55 1.98 0.31 3 0.39 0.30 1.23 0.41 5 0.27 0.25 0.86 0.41

10 0.15 0.18 0.45 0.33 15 0.11 0.14 0.36 0.28 20 0.07 0.11 0.23 0.22 25 0.06 0.10 0.10 0.16 30 0.05 0.09 0.05 0.12

(a) 1.5 m above the top slab (b) 2.5 m below the bottom slab

Fig. 8 Excessive pore pressures in soil foundation.

Transient zone Fully liquefied Transient zone Fully liquefied


The overall rigid body motion of the structure is hardly affected no matter how nonlinearly the under-ground ducts may behave. On the contrary, the averaged shear deformation of the linear duct is much smaller than that of the nonlinear case as demonstrated in Fig. 12. RC nonlinearities cause the duct to accommodate more shear deformation. Besides, as the linear duct re-mains stiff during the earthquake motion, the adjacent unsaturated dry soil movement is restricted by the struc-

ture, and the shear stiffness degradation of the soil is limited (Huo et al. 2005). In addition, the nominal shear stress of the center column, which has much to do with the axial load carrying capacity, demonstrates higher magnitude for the case of nonlinearity rather than the linear case of the structure as shown in Fig. 13. It can be said that nonlinear interaction analysis is indispensable for safety design of underground structures.

(a) Dry sandy ground (b) Saturated sandy ground

(c) With underground structure

Fig. 9 Acceleration response of the ground surface.

(a) Drained condition

(b) Undrained condition

Fig. 10 Horizontal displacement responses of the linear and nonlinear ducts.


(a) Drained condition (b) Undrained condition

(c) Uplift mode of deformation at soil liquefaction (5 times magnified displacement of analysis)

(d) Experimental uplift mode of deformation by Towhata (2008).

Fig. 11 Vertical displacement responses of the linear and nonlinear ducts.

(a) Drained condition – Linear structure (b) Drained condition – Nonlinear structure

(c) Undrained condition – Linear structure (d) Undrained condition – Nonlinear structure

Fig. 12 Shear deformation responses of the linear and nonlinear ducts.


4.3 Yield strength of reinforcement Cracking of concrete and plasticity of steel reinforce-ment are two major sources of RC nonlinearities. In-creased yield strength of reinforcement would reduce nonlinearity of RC elements without varying initial stiffness. Figure 14 and Fig. 15 show the horizontal and vertical displacements of the duct respectively accord-ing to the higher yield strength of 400 MPa. By compar-ing these with the case of 240 MPa (Figs. 10 and 11), it

can be understood that the rigid body motion of the structure is not much affected by the steel nonlinearity. However, the high strength reinforcement would lead to slightly reduced shear response of the duct as shown in Fig. 16. By minimizing the nonlinearity of reinforce-ment, the duct response would come closer to the linear one (Fig. 12(a, c)) although the duct still suffers from larger shear deformations especially in the dry soil.

Figure 17 illustrates the crack pattern of the structure

(a) Drained condition – Linear structure (b) Drained condition – Nonlinear structure

(c) Undrained condition – Linear structure (d) Undrained condition – Nonlinear structure

Fig. 13 Nominal shear stress response of the center column for linear and nonlinear cases.


Fig. 14 Horizontal displacement response of the duct with increased yield strength of rebars.


Fig. 15 Vertical displacement response of the duct with increased yield strength of rebars.


with increased yield strength at the end of dynamic analysis. It can be observed that large cracks are con-centrated at the corners and both ends of the column. Since most of the major cracks occur at the early stages when the acceleration is large and the soil was not yet fully liquefied, the crack pattern in the dry and saturated soil is almost similar. There are small additional cracks at the internal side of the floor slab for the case of lique-faction, because of the “squeezing” pressure of liquefied soil beneath the structure to push it upward. This mechanism is discussed in (Koseki et al. 1997a; Ha-shash et al. 2001; Liu and Song 2006) as the main cause for the uplift of underground structures.

4.4 Effect of Reinforcement ratio The main reinforcement ratio is a key parameter to con-trol the performance of structures practically at the de-sign stage. Thus, parametric analyses with different re-inforcement ratios (Table 4) are performed to investi-gate the responses and damages of the structure associ-ated with the ductility and the shear capacity. The hori-zontal and vertical displacement responses are not influ-enced by the amount of longitudinal reinforcement as shown in Fig. 18 and Fig. 19.

By comparing Fig. 20 with Fig. 12(b, d), it can be said that the imposed structural shear deformation from soil foundation is reduced to about 70% of the standard case by placing two time large amount of reinforcement, and also increases about 20% by placing the half amount of reinforcement. In view of the structural dam-age control, the effectiveness of the reinforcement ratio is found to be comparable no matter how much likely the soil liquefaction occurs. On the other hand, the duc-

tility of the column compression members can be sim-ply estimated (JSCE 2002) by,

u

u

MHV ⋅

=φ (11)

where Vu is the shear capacity of the RC member, H is the shear span (half of the column height in the case of RC duct), Mu is the flexural capacity. When the ductility index φ is greater than 1.3~1.5, practically sufficient inelastic deformation is realized. If it is less than unity, shear failure may take place before yielding of main reinforcement and brittle failure will be initiated. As the main reinforcement ratio is almost proportional to Mu, it must be noted that the increased main reinforcement may decline the RC ductility as well. In considering the sensitivity of reinforcement ratio on the structural duc-tility (output) and imposed deformation (input) from soil foundation, generally, heavy reinforcement is thought not to be effective.

Figure 21 shows the effect of soil stiffness on the maximum shear deformation response of the duct in


Fig. 16 Shear deformation response of the duct with increased yield strength of rebars.


Fig. 17 Crack pattern of the duct after the earthquake motion.

Table 4 Variation of reinforcement ratio for different cases.

Reinforcement ratio Standard Case B Case C

Slab & Wall 1.10% 2.20% 0.55%Long. reinforcement Column 1.60% 3.20% 0.80%

Web reinforcement All elements 0.20% 0.20% 0.20%


Case B and Case C (Table 4). By comparing Fig. 21(a, b) with Fig. 7(a, b), it can be understood that the re-duced amount of main reinforcement may greatly in-crease the ductility demand of the duct especially in the softer layers of soil deposit which structural properties become more effective and predominant. Liquefaction, however, deteriorates the surrounding soil stiffness and thus the deformational demand on the RC duct conse-quently decreases as shown in Fig. 21(c, d). The maxi-

mum shear deformation of the duct in the case C in-creases when the soil is extremely soft (N-value = 1) and full saturated as shown in Fig. 21(d), which is in contrast with Figs. 21(c) and 7(b). It is due to high plas-ticity of yielded reinforcement having low amount of reinforcement in the case C before soil is fully liquefied. However, it is still smaller than that of the same duct under the drained condition.

If this highly nonlinear interaction would not be con-


Fig. 18 Horizontal displacement response of the heavily reinforced duct.


Fig. 19 Vertical displacement response of the heavily reinforced duct.

(a) Drained condition – Case B (b) Drained condition – Case C

(c) Undrained condition – Case B (d) Undrained condition – Case C

Fig. 20 Shear deformation responses of the ducts with different reinforcement ratios.


sidered for the risk of liquefaction, over-reinforcement of web steel, which currently matters in practice, is not unavoidable. 4.5 Effect of Input Excitation In order to investigate the characteristics of input seis-mic excitations, another earthquake record is tried. The north-south component recorded at Hachinohe City dur-ing the Tokachi-oki earthquake of 1968, which is known to have its elongated period, is selected and scaled to 0.3g as shown in Fig. 22. Then, the same procedure which was applied in section 4.1 is followed with the Hachinohe record. The results are shown in Fig. 23, which can be compared by Fig. 7. Besides, in order to make the discussion more clear, the relative shear de-formation is employed, which is defined as the maxi-mum shear deformation of the structure normalized by the maximum distortion of free-field ground response

and can be expressed (Wang 1993) as,

fieldfree

structure

−

=γγ

R (12)

where γ is the maximum angular drift of the rectangular RC duct. The relative shear deformation of the duct in drained and undrained conditions is given in Table 5 and can be compared for different input motions. By contrast, the conclusion made in section 4.1 is found to be valid for different kinds of earthquake properties as long as the surrounding soil fully liquefies. 5. Conclusions

Strong coupling effect of highly nonlinear liquefiable soil and inelastic underground RC was investigated, and consequent damage induced to reinforced concrete was

(a) Drained condition – Case B (b) Drained condition – Case C

(c) Undrained condition – Case B (d) Undrained condition – Case C

Fig. 21 Effect of duct reinforcement ratio on the maximum response of averaged shear deformation: Case B – Heavily reinforced, Case C – Lightly reinforced

Fig. 22 Hachinohe record and its response spectrum.


focused in consideration of rising pore water pressure and the drastic reduction of shear stiffness of soil parti-cle skeleton. The parametric study brings about the fol-lowing conclusions in practice. 1) In case of the drained state of soil foundation with-

out liquefaction, the smaller stiffness of soil layer is, the larger ductility of the underground RC duct is demanded in structural design. Consequently, large amount of web reinforcement is being specified in practical design.

2) In case of the liquefiable soil, however, the structural damage is found to be reduced owing to the signifi-cantly reduced shear stiffness of soil foundation, even though the larger soil deformation is realized. Thus, the ductility demand to the structure is sharply lighten due to the effect of soil seismic isolation un-der very loosely deposited foundation. If this high nonlinearity would not be considered in practice, so heavy reinforcement is inevitably forced which causes difficulty of concreting works. Of course, it should be pointed out that some countermeasures such as sheet piling and controlling the weight of tunnels should be considered to reduce the rigid body motion of underground structures in liquefiable soils as being conducted in practice. The shear de-formation and the uplift are in relation of trade-off as simulated in this study.

3) The uplift rigid body motion caused by liquefaction is hardly affected by the nonlinearity of underground RC. In other words, the uplift stability cannot be controlled well by the structural design except for self-weight. In fact, lots of underground ducts were reported to drift upwards without accompanying any structural damage. These past experiences are con-firmed by the nonlinear coupling analysis as well.

4) Reinforcement with higher yield strength may re-duce the deformation and the damage to RC ducts. But its effectiveness is practically negligibly small.

5) Increased main reinforcement may reduce the mean deformation and material damages to the under-ground ducts with similar magnitude for both drained and undrained states of soil foundation. If the soil is loosely deposited with high risk of lique-faction, the structural damage may be less than the

case of unsaturated dry soil.

References An, X., Shawky, A. A. and Maekawa, K. (1997). “The

collapse mechanism of a subway station during the Great Hanshin earthquake.” Cement and Concrete Composite, 19, 241-257.

Biot, M. A. (1962). “Mechanics of deformation and acoustic propagation in porous media.” Journal of Applied Physics, 33(4), 1482-1498.

Hall, W. J. and O’Rourke, T. D. (1991). “Seismic behavior and vulnerability of pipelines.” Proc. Third US Conf. Lifeline Earthquake Engineering, 761-773.

Hashash, Y. M. A., Hook, J. J., Schmidt, B. and Yao, J. (2001). “Seismic design and analysis of underground structures.” Tunneling Underground Space Technology, 16(4), 247-293.

Huo, H., Bobet, A., Fernandez, G. and Ramirez, J. (2005). “Load transfer mechanisms between underground structure and surrounding ground: Evaluation of the failure of the Daikai station.” Journal of Geoenvironmental Engineering, 131(12), 1522-1533.


Fig. 23 Effect of duct reinforcement ratio on the maximum response of averaged shear deformation.

Table 5 The relative shear deformation of the duct in different cases.

Relative shear deformation (%)

Kobe Record Hachinohe Record SPT N-

valueDrained Undrained Drained Undrained

1 0.46 0.16 0.69 0.19 3 0.77 0.33 0.93 0.26 5 0.94 0.48 0.78 0.38

10 1.20 0.74 1.07 0.64 15 1.29 0.77 1.48 0.78 20 1.53 0.98 1.71 0.86 25 1.67 1.56 1.43 1.12 30 1.85 2.54 1.52 1.55


Japan Society of Civil Engineers (2002). “Recommended code and manual for seismic performance verification of out-door important structures of nuclear power plants.” Concrete Library International.

Kato, B. (1979). “Mechanical properties of steel under load cycles idealizing seismic action.” CEB Bulletin D’Information, 131, 7-27.

Kimura, T., Takemura, J., Hiro-oka, A. and Okamura, M. (1995). “Countermeasures against liquefaction of sand deposits with structures.” Proc. First International Conference on Earthquake Geotechnical Engineering, 1203-1224, Tokyo, Japan.

Koseki, J., Matsuo, O., Koga, Y. (1997a). “Uplift behavior of underground structures caused by liquefaction of surrounding soil during earthquake.” Soils and Foundations, 37(1), 97-108.

Koseki, J., Matsuo, O., Ninomiya, Y., Yoshida, T. (1997b). “Uplift of sewer manholes during the 1993 Kushiro-Oki earthquake.” Soils and Foundations, 37(1), 109-121.

Li, B., Maekawa, K. and Okamura, H. (1989). “Contact density model for stress transfer across crack in concrete.” Journal of Faculty of Engineering, University of Tokyo (B), 40(1), 9-52.

Liu, H. and Song, E. (2005). “Seismic Response of Large Underground Structures in Liquefiable Soils Subjected to Horizontal and Vertical Earthquake Excitations.” Computers Geotechnics, 32, 223-244.

Liu, H., Song, E. (2006). “Working mechanism of cutoff walls in reducing uplift of large underground structures induced by soil liquefaction.” Computers Geotechnics, 33, 209-221.

Maekawa, K., Irawan, P. and Okamura, H. (1997). “Path-dependent three-dimensional constitutive laws of reinforced concrete: Formulation and experimental verifications.” Structural Engineering and Mechanics, 5(6), 743-54.

Maekawa, K. and An, X. (2000). “Shear failure and ductility of RC columns after yielding of main reinforcement.” Engineering Fracture Mechanics, 65, 335-368.

Maekawa, K., Pimanmas, A. and Okamura, H. (2003). “Nonlinear Mechanics of Reinforced Concrete.” Spon Press, London.

Maekawa, K., Fukuura, N. and Soltani, M. (2008). “Path-dependent high cycle fatigue modeling of joint interfaces in structural concrete.” Journal of Advanced Concrete Technology, 6(1), 227-242.

Maki, T. and Mutsuyoshi, K. (2004). “Seismic behavior of reinforced concrete pile under ground.” Journal of Advanced Concrete Technology, 2(1), 37-47.

Maki, T., Maekawa, K. and Mutsuyoshi, H. (2005). “RC pile-soil interaction analysis using a 3D-finite element method with fiber theory-based beam elements.” Earthquake Engineering and Structural Dynamics, 99, 1-26.

Masing, G. (1926). “Eigenspannungen and Verfestigung Beim Messing.” Proc. of Second International Congress of Applied Mechanics, 332, Zurich.

Nam S.H., Song H.W., Byun K.J. and Maekawa K. (2006). “Seismic analysis of underground reinforced concrete structures considering elasto-plastic interface element with thickness.” Engineering Structures, 28, 1122-1131.

O’Rourke, T. D., Goh, S. H., Menkiti, C. O. and Mair, R. J. (2001). “Highway tunnel performance during the 1999 Duzce earthquake.” Proc. 15th International Conference on Soil Mechanics and Geotechnical Engineering.

Tamari, Y. and Towhata, I. (2003). “Seismic soil-structure interaction of cross sections of flexible underground structures subjected to soil liquefaction.” Soils and Foundations, 43(2), 69-87.

Towhata, I. and Ishihara, K. (1985). “Modeling soil behaviors under principal stress axes rotation.” 5th Int. Conf. on Numerical Method in Geomechanics, Nagoya, 523-530.

Towhata, I. (2008). “Geotechnical Earthquake Engineering.” Springer, Germany.

Wang, J. N. (1993). “Seismic design of tunnels: a state-of-the-art approach.” Monograph, monograph 7, Parsons, Brinckerhoff, Quade and Douglas Inc, New York.

Wang, W. L. , Wang, T. T. , Su, J. J., Lin, C. H., Seng, C. R. and Huang, T. H. (2001). “Assessment of damage in mountain tunnels due to the Taiwan Chi-Chi earthquake.” Tunneling Underground Space Technology, 16(3), 133-150.

nonlinear seismic response and damage of reinforced concrete

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