Fully Nonlinear versus Equivalent Linear Computation Method for ...

Fully Nonlinear versus Equivalent Linear Computation Method for Seismic Analysis of Midrise Buildings on Soft Soils Downloaded from ascelibrary.org by HONG KONG POLYTECHNIC UNIV on 03/27/14. Copyright ASCE. For personal use only; all rights reserved.

Behzad Fatahi1 and S. Hamid Reza Tabatabaiefar2 Abstract: In this study, the accuracy of a fully nonlinear method against an equivalent linear method for dynamic analysis of soil-structure interaction is investigated comparing the predicted results of both numerical procedures. Three structural models, including 5-story, 10-story, and 15-story buildings, are simulated in conjunction with two soil types with shear-wave velocities less than 600 m=s. The aforementioned frames were analyzed under three different conditions: (1) fixed-base model performing conventional time history dynamic analysis under the influence of earthquake records, (2) flexible-base model (considering full soil-structure interaction) conducting equivalent linear dynamic analysis of soil-structure interaction under seismic loads, and (3) flexible-base model performing fully nonlinear dynamic analysis of soilstructure interaction under the influence of earthquake records. The results of these three cases in terms of average lateral story deflections and interstory drifts are determined, compared, and discussed. It is concluded that the equivalent linear method of the dynamic analysis underestimates the inelastic seismic response of midrise moment resisting building frames resting on soft soils in comparison with the fully nonlinear dynamic analysis method. Therefore, a design procedure using the equivalent linear method cannot adequately guarantee the structural safety for midrise building frames resting on soft soils. DOI: 10.1061/(ASCE)GM.1943-5622.0000354. © 2014 American Society of Civil Engineers. Author keywords: Inelastic seismic response; Equivalent linear method; Fully nonlinear method; Inelastic design procedure.

Introduction The problem of soil-structure interaction (SSI) in the seismic analysis and design of structures has become increasingly important because it may be inevitable to build structures at locations with less favorable geotechnical conditions in seismically active regions. The December 28, 1989, Newcastle (Australia) Earthquake killed and injured over 150 people, and the damage bill was about $4 billion. Recently, a similar disaster hit Haiti on January 12, 2010, causing over 200,000 deaths and left over 3 million people homeless. The similarity of these two earthquakes is that both of them are intraplate earthquakes occurring in the interior of a tectonic plate. In both cases, many midrise buildings (approximately 5–15 stories) were severely damaged. In addition, the scarcity of land compels engineers to construct major structures over soft deposits. Therefore, there is a need to design structures safely, but not costly, against natural disasters, such as earthquakes. Evidently, the effects of dynamic SSI under extreme loads caused by strong earthquakes are significant for many classes of structures and must be included precisely in the design. Over the last few years, application of the concept of performance-based seismic design has been promoted and developed rapidly. The development of this approach has been a natural outgrowth of the evaluation and upgrade process for existing 1 Senior Lecturer of Geotechnical Engineering, Centre for Built Infrastructure Research, Univ. of Technology Sydney, Sydney, NSW 2007, Australia. E-mail: [email protected] 2 Research Assistant, Centre for Built Infrastructure Research, Univ. of Technology Sydney, Sydney, NSW 2007, Australia (corresponding author). E-mail: [email protected] Note. This manuscript was submitted on December 9, 2012; approved on August 28, 2013; published online on August 30, 2013. Discussion period open until August 27, 2014; separate discussions must be submitted for individual papers. This paper is part of the International Journal of Geomechanics, © ASCE, ISSN 1532-3641/04014016(15)/$25.00.

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buildings. Performance objectives are expressed as an acceptable level of damage, which are typically categorized as one of several performance levels. Performance levels describe the state of structures after being subjected to a certain hazard level and are classified as fully operational, operational, life safe, near collapse, or collapse [Applied Technology Council (ATC) 1997]. Overall lateral deflection, ductility demand, and interstory drifts are the most commonly used damage parameters. The aforementioned five qualitative performance levels are related to the corresponding quantitative maximum interstory drifts of 0.2, 0.5, 1.5, 2.5, and .2:5%, respectively. Several researchers (Spyrakos et al. 1989; Veletsos and Prasad 1989; Safak 1995; Krawinkler et al. 2003; Galal and Naimi 2008; El Ganainy and El Naggar 2009; Iida 2012; Tabatabaiefar et al. 2012, 2013) studied structural behavior of unbraced structures subjected to earthquake under the influence of SSI. In addition, during recent decades, the importance of dynamic SSI for several structures founded on soft soils has been well recognized. Examples are given by Gazetas and Mylonakis (1998), including evidence that some structures founded on soft soils are vulnerable to SSI. Several efforts have been made in recent years in the development of numerical methods for assessing the response of structures and supporting soil media under seismic loading conditions. Successful application of these methods for determining ground seismic response is vitally dependent on the incorporation of the soil properties in the analyses. As a result, substantial effort has also been made toward the determination of soil attributes for use in these analytical procedures. There are two main numerical procedures for computation of the seismic response of structures under the influence of SSI: equivalent linear method and fully nonlinear method. The traditional standard practice for dynamic analysis of soilstructure systems has been based on the equivalent linear method. Various analytical studies have been carried out to compute the seismic response of structures adopting equivalent linear dynamic analysis for SSI (Stewart et al. 1999; Kim and Roesset 2004;

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Dutta et al. 2004; Tabatabaiefar and Massumi 2010; Maheshwari and Sarkar 2011) because of its simplicity and adoptability to most structural software around the globe. However, the effects of soil nonlinearity of supporting soil on the seismic response of structures have not been fully addressed in the literature using the fully nonlinear dynamic analysis method. The fully nonlinear analysis has not been applied as often in practical design as a result of its complexity and the requirement for advanced computer programs. However, practical applications of the fully nonlinear analysis have increased in the last decade as more emphasis is placed on reliable predictions in dynamic analysis of complex soil-structure systems (Byrne et al. 2006). In this study, the accuracy of the fully nonlinear method assessed against the equivalent linear method for dynamic analysis of SSI is investigated by comparing the predicted results of both numerical procedures. The main goal of this comparison is to pinpoint whether the simplified equivalent linear method of analysis is adequately accurate to reliably determine the inelastic seismic response of midrise building frames or if it is necessary to use the fully nonlinear method to attain rigorous and reliable results.

Numerical Procedures for Computation of Seismic Response The equivalent linear method has been in use for many years to compute the seismic response of the structures at sites subjected to seismic excitation. In the equivalent linear method, a linear analysis is carried out with some assumed initial values for damping and shear modulus ratios of the model, which are often referred to as equivalent linear material parameters. Then, the maximum cyclic shear strain is recorded for each element and used to determine the new values for the damping and modulus, using the backbone curves relating the damping ratio and secant modulus to the amplitude of the shear strain. The new values of the damping ratio and shear modulus are then used in the next stage of the numerical analysis. The whole process is repeated several times until there is no further change in the properties and structural response. At this stage, straincompatible values of the damping and modulus are recorded, and the simulation using these values is deemed to be the best possible prediction of the real behavior. Rayleigh damping may be used in this method to simulate energy losses in the soil-structure system when subjected to dynamic loading. Seed and Idriss (1969) noted that the equivalent linear method uses linear properties for each element, which remain constant under the influence of seismic excitations. Those values, as explained, are estimated from the mean level of dynamic motion. Other characteristics of the equivalent linear method are as follows (Seed and Idriss 1969): • The interference and mixing phenomena taking place between different frequency components in a nonlinear material are missing from an equivalent linear analysis; • The method does not directly provide information on irreversible displacements and permanent changes; and • In the case where both shear and compression waves are propagated through a site, the equivalent linear method typically treats these motions independently. Despite the aforementioned drawbacks of the method, because most structural software can only adopt the equivalent linear method, it has been considered as a simple and popular approach for dynamic analysis of SSI. The fully nonlinear method is capable of precisely modeling nonlinearity in the dynamic analysis of soil-structure systems and follows any prescribed nonlinear constitutive relationship. In addition, structural geometric nonlinearities (large displacements) can be © ASCE

precisely accommodated in this method. During the solution process, structural materials could behave as isotropic, linearly elastic materials with no failure limit for an elastic analysis or as elastoplastic materials with a specified limiting plastic moment for inelastic structural analysis to simulate elastic-perfectly plastic behavior. For the dynamic analysis, the damping of the system in the numerical simulation should be reproduced in magnitude and form, simulating the energy losses in the natural system subjected to the dynamic loading. In soil and rock, natural damping is mainly hysteretic (Gemant and Jackson 1937). The hysteretic damping algorithm, which is incorporated in this solution method, enables the strain-dependent modulus and damping functions to be incorporated directly into the numerical simulation. Other characteristics of the fully nonlinear method are as follows (Byrne et al. 2006): • Nonlinear material law, interference, and a combination of different frequency components can be considered simultaneously; • Irreversible displacements and other permanent changes can be modeled as required; and • Both shear and compression waves are propagated together in a single simulation, and the material responds to the combined effect of both components. To perform a fully nonlinear dynamic analysis, a computer program treating soil nonlinearity and structural inelastic behavior is required. Therefore, structural engineers prefer to use the equivalent linear approach using a trial and error process based on the available seismic code recommendations. Byrne et al. (2006) and Beaty and Byrne (2001) reviewed these methods and discussed the benefits of the fully nonlinear numerical method over the equivalent linear method for different practical applications. The equivalent linear method does not directly capture any nonlinearity effects caused by its linear solution process. In addition, strain-dependent modulus and damping functions are only taken into account in an average sense to approximate some effects of nonlinearity, whereas the fully nonlinear method correctly represents the physics associated with the problem and follows any stress-strain relationship in a realistic way. In this method, the smallstrain shear modulus and damping degradation of soil with a strain level can be precisely captured in the modeling.

Developed Numerical Soil-Structure Model The governing equations of motion for the structure incorporating foundation interaction and the method of solving these equations are relatively complex. Therefore, the direct method, a method in which the entire soil-structure system is modeled in a single step, is used. The soil-structure system simulated adopting the direct method, composed of structure, common nodes, soil foundation system, and earthquake-induced acceleration at the level of the bedrock, is shown in Fig. 1. The dynamic equation of motion of the soil and structure system can be written as ½Mf€ ug þ ½Cfug _ þ ½Kfug ¼ 2½Mfmg€ ug þ fFv g

(1)

_ and f€ where fug, fug, ug 5 nodal displacements, velocities, and accelerations with respect to the underlying soil foundation, respectively; and ½M, ½C, and ½K 5 mass, damping, and stiffness matrices of the structure, respectively. It is more appropriate to use the incremental form of Eq. (1) when plasticity is included; then, the matrix ½K should be the tangential matrix, and f€ ug g is the earthquake-induced acceleration at the level of the bedrock. For example, if only the horizontal acceleration is considered, then fmg 5 ½1, 0, 1, 0, . . . , 0:1, 0T . fFv g is the force vector corresponding to the viscous boundaries. This vector is nonzero only when there is

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a difference between the motion on the near side of the artificial boundary and the motion in the free field (Wolf 1998). To model a soil-structure system in the direct method, a novel and enhanced soil-structure model is developed in Fast Lagrangian Analysis of Continua (FLAC2D) software (Itasca Consulting Group 2008) to simulate various aspects of complex dynamic SSIs in a realistic and rigorous manner. The FLAC2D software is a twodimensional (2D) explicit finite-difference program for engineering mechanics computations. This program can simulate the behavior of different types of structures. Materials are represented by elements, which can be adjusted to fit the geometry of the model. Each element behaves according to a prescribed linear or nonlinear stress/strain

Fig. 1. Soil-structure system in the direct method

law in response to the applied forces or boundary restraints. The FLAC2D software uses an explicit, time marching method to solve the algebraic equations. The basic formulation for FLAC2D is for a 2D plane-strain model associated with long structures with invariable cross sections with the external loads acting in the plane of the cross sections. The dynamic equations of motion are included in the FLAC2D formulation. In the general solution procedure of the dynamic equations of motion, FLAC2D invokes the equations of motion to derive new velocities and displacements from stresses and forces. Then, strain rates are derived from velocities, and new stresses are obtained from the strain rates. The soil-structure model, illustrated in Fig. 2, uses beam structural elements to model beams, columns, and foundation slabs. The 2D plane-strain grids composed of quadrilateral elements are used to model the soil medium. Nonlinear behavior of the soil medium (in the fully nonlinear dynamic analysis case) can be captured using backbone curves of the shear modulus ratio versus shear strain (G=Gmax 2 g) and the damping ratio versus shear strain (j 2 g) adopting the Mohr-Coulomb failure model. The foundation facing zone in the numerical simulations is separated from the adjacent soil zone by interface elements to simulate frictional contact. The interface between the foundation and soil is represented by normal (kn ) and shear (ks ) springs between two planes contacting each other and is modeled using a linear spring system with the interface shear strength defined by the Mohr-Coulomb failure criterion (Fig. 3). The relative interface movement is controlled by interface stiffness values in the normal and tangential directions. Normal and shear spring stiffness values for interface elements of the soil-structure model are set to 10 times the equivalent stiffness of the neighboring zone based on the relationship recommended by Rayhani and El Naggar (2008) and the Itasca Consulting Group (2008) for the isotropic soil medium as follows:

Fig. 2. Components of the soil-structure model

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Fig. 3. Interface elements including normal and shear stiffness springs

Fig. 4. Simulating lateral boundary conditions for the soil-structure model

0 1 K þ 4G B 3 C ks ¼ kn ¼ 10@ A Dzmin

(2)

where K and G 5 bulk and shear modulus of the neighboring zone, respectively; and Dzmin 5 smallest width of an adjoining zone in the normal direction. The shear strength of the interface has been simulated using the Mohr-Column model, assuming that there is no slip in the interface until the shear strength of the interface is reached; after that, perfectly plastic deformation would occur as a result of the associate plastic flow rule being adopted. The normal and shear springs control the force transfer process from one surface to the other. This is a simplifying assumption that has only been used for interface modeling. In this study, Dzmin is equal to 1 because the soil mesh size has been considered to be 1 3 1 m for both the foundation and soil. By recording and comparing the shear (horizontal) displacements at the base of the foundation and the soil surface (two sides of interface), there has been insignificant slip between the soil and foundation. For lateral boundaries of the soil medium, quiet boundaries (viscous boundaries), proposed and developed by Lysmer and Kuhlemeyer (1969), are used in this study. The proposed method is based on the use of independent dashpots in the normal and shear directions at the model boundaries. The dashpots provide viscous normal and shear tractions given by Tn ¼ 2r  CP  nn

(3)

Ts ¼ 2r  Cs  vs

(4)

where Tn and Ts 5 normal and shear tractions at the model boundaries, respectively; vn and vs 5 normal and shear components © ASCE

Fig. 5. Used earthquake acceleration record for this study: (a) near-field acceleration record of the Kobe earthquake (1995); (b) near-field acceleration record of the Northridge earthquake (1994); (c) far-field acceleration record of the El Centro earthquake (1940); (d) far-field acceleration record of the Hachinohe earthquake (1968)

of the velocity at the boundary, respectively; r 5 material density; and Cp and Cs 5 p-wave and s-wave velocities, respectively. The lateral boundaries of the main grid are coupled to the freefield grids by viscous dashpots of quiet boundaries at the sides of the model to simulate the free-field motion, which would exist in the absence of the structure. Fig. 4 shows the simulated lateral boundary condition for the developed soil-structure model. To perform a comprehensive investigation on the seismic response of the structure models, two near-field earthquake acceleration records, including Kobe in 1995 [Fig. 5(a)] and Northridge in 1994 [Fig. 5(b)] and two far-field earthquake acceleration records comprising El Centro in 1940 [Fig. 5(c)] and Hachinohe in 1968 [Fig. 5(d)], were selected and used in the time history analysis. These earthquakes have been chosen by the International Association for

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Structural Control and Monitoring for benchmark seismic studies (Karamodin and Kazemi 2008). The characteristics of the earthquake ground motions are provided in Table 1. The proposed numerical soil-structure model has been verified and validated by performing experimental shaking table tests at the University of Technology Sydney Civil Laboratories. For this purpose, a prototype soil-structure system has been designed and constructed to realistically simulate the free-field conditions in shaking table tests. A series of shaking table tests were performed on the soilstructure physical model under the influence of four scaled earthquake acceleration records, and the nonlinear results were measured and compared with the numerical predictions. In comparing the predicted and observed values, the numerical predictions and laboratory measurements are in a good agreement. Therefore, the numerical soil-structure model can replicate the behavior of the real soil-structure system with acceptable accuracy (Tabatabaiefar 2012). In addition, no reflections of outward propagating waves back to the model were recorded in numerical predictions and laboratory measurements.

Characteristics of Structural Models According to Chandler et al. (2010), midrise buildings are an aggregation of dwelling buildings ranging from 5 to 15 stories. With respect to this definition, to cover this range, three structural models consisting of 5-story, 10-story, and 15-story models, representing conventional types of midrise RC moment resisting building frames, have been selected in this study as per the specifications provided in Table 2. The selected span width conforms to architectural norms and construction practices of the conventional buildings in megacities. For the structural concrete used in this analysis and design, specified compressive strength ( fc9) and mass density ( r) are assumed to be 32 MPa and 2,400 kg=m3 , respectively. The modulus of elasticity of concrete (E) was calculated according to Clause 3.1.2.a of AS3600 (Australian Standard 2009) as follows:  pffiffiffiffi E ¼ ðrÞ1:5  0:043 fc9

(5)

In this study, structural sections of the models are designed based on an inelastic method assuming elastic-perfectly plastic behavior for the structural members. For this purpose, structural members of the models (Table 1) are simulated in SAP2000 14 software reflecting various geometries and properties of Models S5 (5 stories), S10 (10 stories), and S15 (15 stories). Then, gravity loads, including permanent (dead) and imposed (live) actions, are determined and Table 1. Earthquake Ground Motions Used Earthquake

Country

Northridge Kobe El Centro Hachinohe

United States Japan United States Japan

T (s) Year PGA (g) MW (R) duration 1994 1995 1940 1968

0.843 0.833 0.349 0.229

6.7 6.8 6.9 7.5

30.0 56.0 56.5 36.0

Type Near field Near field Far field Far field

applied to the structural models in accordance with AS/NZS1170.1 (Australian Standards 2002) (permanent, imposed, and other actions). The values of permanent and imposed actions are determined as uniform distributed loads over the floors according to AS/NZS1170.1, considering the spacing of the frames being 4 m, permanent action (G) equal to 6 kPa, and imposed action (Q) equal to 2 kPa. Because the studied concrete frames are moment resisting building frames, beam-column connections are fixed connections enabling both forces and moments to transfer. In the studied moment resisting frames, the connections of columns to the concrete foundation are fixed as well. Inelastic time history dynamic analyses under the influence of four earthquake ground motions, provided in Table 1, were then performed on the structural models. The generic process of the inelastic analysis is similar to a conventional elastic procedure. The primary difference is that the properties of the components of the model include the plastic moment in addition to the initial elastic properties. These are normally based on approximations derived from test results on individual components or theoretical analyses (ATC 1996). In this study, inelastic bending is simulated in the structural elements by specifying a limiting plastic moment. When the plastic moment is specified, the value may be calculated by considering a flexural structural member of width b and height h with yield stress sy . If the member is composed of a material that behaves in an elastic-perfectly plastic manner (Fig. 6), the plastic resisting moments (M P ) for the rectangular sections can be computed as follows:   bh2 (6) M P ¼ sy 4 Present formulations, adopted for inelastic analysis and design, assume that structural elements behave elastically until reaching the defined plastic moment. The section at which the plastic moment (M P ) is reached can continue to deform without inducing additional resistance. Plastic moments, (M P ), for each concrete section of Models S5, S10, and S15 have been determined according to Eq. (6) and assigned to the sections considering the yield stress of concrete material (sy ) equal to the compressive strength of concrete ( fc9). In addition, geometric nonlinearity and P-Delta effects are considered according to AS3600 (Australian Standard 2009), and cracked sections for the RC sections are taken into consideration by multiplying the cracked section coefficients by the stiffness values of the structural members (EI) according to ACI 318 (American Concrete Institute 2008). Based on this standard, cracked section coefficients are 0.35 and 0.70 for beams and columns, respectively. Afterward, structural members were designed in accordance with AS3600 (Australian Standard 2009) in a way that performance levels of the designed models stay in the life safe level by limiting the maximum inelastic interstory drifts to 1.5%. To determine inelastic interstory drifts for each two adjacent stories, the maximum lateral deflection of each story is derived from SAP2000 deflection history records. Using the maximum story deflections, inelastic interstory drifts have been determined using the following equation based on AS1170.4 (Australian Standards 2007):

Table 2. Dimensional Characteristics of Studied Frames Reference name (code) Number of stories Number of bays Story height (m) Bay width (m) Total height (m) Total width (m) S5 S10 S15

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5 10 15

3 3 3

3 3 3

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4 4 4

15 30 45

12 12 12

Spacing of the frames into the page (m) 4 4 4

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drift ¼ ðdiþ1 2 di Þ=h

(7)

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where di11 5 deflection at the (i 1 1) level; di 5 deflection at the (i) level; and h 5 story height. In practical designs, it is often assumed that the story deflection is equal to the horizontal displacement of the nodes on the level, which may be caused by translation, rotation, and distortion. In the final selection of the beam and column sections, constructability and norms have been considered.

Properties of Soils Used

Fig. 6. Elastic-perfectly plastic behavior of structural elements

According to the available literature, generally when the shear-wave velocity of the supporting soil is less than 600 m=s, the effects of the SSI on the seismic response of structural systems, particularly for moment resisting building frames, are significant (Veletsos and Meek 1974; Galal and Naimi 2008; Nateghi Rezaei-Tabriz 2011; Tabatabaiefar et al. 2012, 2013). Therefore, in this study, two relatively soft clayey soils with shear-wave velocity less that 600 m=s, representing soil classes De and Ee according to AS1170.4 (Australian Standards 2007), have been used. Characteristics of the adopted soils are provided in Table 3. The subsoil properties have

Table 3. Geotechnical Characteristics of Adopted Soils Shear-wave velocity Vs (m=s)

Unified Soil Classification System

Maximum shear modulus Gmax (kPa)

Poisson’s ratio

Soil density r (kg=m3 )

c9 (kPa)

f9 (degrees)

De

320

177,304

0.39

1,730

20

19

Ee

150

Inorganic clays of low to medium plasticity (CL) Inorganic clays of low to medium plasticity (CL)

33,100

0.40

1,470

20

12

Soil type (Australian Standards 2007)

Reference Rahvar (2006a) Rahvar (2006b)

Fig. 7. Numerical models: (a) fixed-base model; (b) flexible-base model

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generates a cyclic shear strain of about 1024 % where the resulting shear modulus is called Gmax . In the event of an earthquake, the cyclic shear strain amplitude increases, and the shear strain modulus and damping ratio, which both vary with the cyclic shear strain amplitude, change relatively. The damping and tangent modulus are

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been extracted from actual in situ and laboratory tests (Rahvar 2006a, b). Therefore, these parameters have merits over the assumed parameters, which may not be completely conforming to reality. The shear-wave velocity values, provided in Table 3, have been obtained from downhole tests, which are low-strain in situ tests. This test

Fig. 8. Adopted fitting curves for clay: (a) relationships between G=Gmax versus cyclic shear strain; (b) relationships between material damping ratio versus cyclic shear strain Table 4. Average Base Shear Ratios of Models S5, S10, and S15 for Three Different Cases Soil type Ee

Soil type De

Case 1: fixed-base model Reference name (code)

V (kN)

Case 2: flexible-base equivalent linear method ~ Eq (kN) V

~ Eq =V V

Case 3: flexible-base fully nonlinear method

Case 2: flexible-base equivalent linear method

Case 3: flexible-base fully nonlinear method

~ Fn (kN) V

~ Eq (kN) V

~ Fn (kN) V

~ Fn =V V

~ Eq =V V

~ Fn =V V

S5 76 64 0.84 61 0.80 58 0.76 50 0.66 S10 225 181 0.80 166 0.73 140 0.62 113 0.50 S15 338 230 0.68 205 0.60 195 0.57 150 0.44 ~ Fn 5 base shear of flexible-base ~ Eq 5 base shear of flexible-base model determined by equivalent linear method; V Note: V 5 base shear of fixed-base model; V model determined by fully nonlinear method. © ASCE

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in agreement with most modern seismic codes (ATC 1996; Building Seismic Safety Council 2003). Those seismic codes evaluate local site effects just based on the properties of the top 30 m of the soil profile. In addition, it is assumed that the water table is below the bedrock level.

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Numerical Simulation and Analysis In this study, to determine the accuracy of the equivalent linear method versus the fully nonlinear method, the main differences between the seismic responses of the midrise moment resisting building frames resting on soft soils, predicted by both numerical procedures, are investigated. For this purpose, three structural models, including 5-story, 10-story, and 15-story buildings (Table 2), are simulated in conjunction with two soil types with shear-wave velocities less than 600 m=s, representing soil classes De and Ee (Table 3) according to the classification of AS1170.4 (Australian Standards 2007) (earthquake actions in Australia), having 30-m bedrock depth. To study the main differences between the results predicted by the equivalent linear method and the fully nonlinear method for dynamic analysis of SSI of the selected building frames, inelastic dynamic time history analyses were carried out using FLAC2D software. In the inelastic dynamic analysis, geometric nonlinearity of the structures capturing P-Delta effects is accommodated by specifying the large-strain solution mode in the FLAC2D software. In addition, cracked sections for the RC sections are taken into account by multiplying the moment of inertia of the uncracked sections (Ig ) by the cracked section coefficients (0:35Ig for beams and 0:70Ig for columns) according to Section 10.11.1 of ACI 318 (American Concrete Institute 2008). Three different cases were investigated in this study as follows. Case 1 Case 1 consists of fixed-base columns on rigid ground and is called the fixed-base model. In this case, inelastic time history dynamic analysis under the influence of the earthquake records, provided in Table 1, including Kobe in 1995 [Fig. 5(a)] and Northridge in 1994 [Fig. 5(b)] and El Centro in 1940 [Fig. 5(c)] and Hachinohe in 1968 [Fig. 5(d)], were performed. The earthquake records were applied to the fixed base of the structural models. Fig. 7(a) illustrates simulated fixed-base models in FLAC2D. The results of inelastic dynamic analyses of Case 1 were considered as pure structural analyses without considering the SSI effects in the analyses. Finally, the results of inelastic time history dynamic analyses in terms of base shears, lateral deflections, and interstory drifts under the influence of the four earthquake ground motions were derived from FLAC2D history records for fixed-base models. Case 2

Fig. 9. Lateral deflections of Model S15 resting on soil class Ee for three different cases under the influence of (a) Kobe earthquake (1995); (b) Northridge earthquake (1994); (c) El-Centro earthquake (1940); (d) Hachinohe earthquake (1940)

selected to be appropriate to the level of excitation at each point in time and space, which is called the hysteretic damping algorithm. The bedrock depth is assumed to be 30 m because the most amplification occurs within the first 30 m of the soil profile, which is © ASCE

The second case is the flexible-base model considering the soil medium underneath the structure, called the soil-structure model. It uses the direct method to model and analyze the dynamic SSI using the equivalent linear method of analysis. To model the soil-structure system in the direct method, the developed soil-structure model, described previously in detail, was used to simulate various aspects of complex dynamic SSIs in a realistic and rigorous manner. The simulated soil-structure models are shown in Fig. 7(b). To perform the equivalent linear analysis, a linear analysis was carried out with assumed initial values for the equivalent linear material parameters, including the damping ratio (j) and shear modulus ratio (G=Gmax ), of the model. For this purpose, with respect

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to the peak ground acceleration (PGA) of the used earthquake records (Table 1), the initial value of the shear modulus ratio (G=Gmax ), equal to 0.42, was derived from Table 10–5 of ATC-40 (ATC 1996). Having the initial value of ðG=Gmax Þ 5 0:42, the initial value of the cyclic shear strain was extracted, equal to 0.3%, from the backbone curve of G=Gmax versus cyclic shear strain for cohesive soils presented by Sun et al. (1998) [Fig. 8(a)]. Afterward, the initial value of the soil damping ratio ðjÞ 5 16% was found from the backbone curve of the damping versus cyclic shear strain for cohesive soils [Fig. 8(b)] presented by Sun et al. (1998). Using the extracted initial values for the equivalent linear material parameters, inelastic time history dynamic analysis under the influence of the four mentioned earthquake records (Table 1) was performed on flexible-base models. The earthquake records were applied to the combination of soil and structure directly at the bedrock level. Then, the maximum soil cyclic shear strain values were determined from FLAC2D outputs. With respect to the extracted maximum soil cyclic shear strain values and using backbone curves of G=Gmax versus cyclic shear strain and damping versus cyclic shear strain for cohesive soils (Fig. 8), the new values of G=Gmax and the damping ratio were

determined. The whole process was repeated several times until achieving the strain-compatible values of the damping ratio (j) and shear modulus ratio (G=Gmax ). Acceptable error for reaching the strain-compatible values of the damping ratio (j) and shear modulus ratio (G=Gmax ) is 5% according to ATC-40 (ATC 1996). As previously mentioned, the simulation using these values is deemed to be the best possible prediction of the real soil-structure system seismic behavior while adopting linearity during the analysis. Eventually, the results of the equivalent linear dynamic analyses using the stain compatible values of the damping ratio (j) and shear modulus ratio (G=Gmax ) in terms of base shears, lateral deflections, and interstory drifts under the influence of the earthquake records were determined from the FLAC2D history records for flexible-base models resting on soil classes De and Ee . Case 3 Case 3 is a flexible-base model to model and analyze dynamic SSIs using the fully nonlinear method of analysis. The soil-structure model is similar to Case 2 in which the developed soil-structure models,

Fig. 10. Average seismic response of Model S5 resting on soil class De for three different cases: (a) lateral deflections; (b) interstory drifts

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presented in Fig. 7(b), were used. By adopting the fully nonlinear method in dynamic analysis, nonlinear behavior of the soil medium has been captured using backbone curves of the shear modulus ratio versus shear strain (G=Gmax 2 g) and damping ratio versus shear strain (j 2 g) adopting the Mohr-Coulomb failure model. The fully nonlinear method is capable of precisely modeling nonlinearity in the dynamic analysis of soil-structure systems and following any prescribed nonlinear constitutive relationship. The fully nonlinear method adopts a hysteretic damping algorithm, which captures the hysteresis curves and energy-absorbing characteristics of the real soil. The small-strain shear modulus and damping degradation of the soil with a strain level can be considered in the modeling accurately. In this study, the tangent modulus function presented by Hardin and Drnevich (1972), known as the Hardin model, is used to implement hysteretic damping to the numerical models. This model is defined by Ms ¼

1 1 þ g=gref

(8)

where Ms 5 secant modulus (G=Gmax ); g 5 cyclic shear strain; and gref 5 Hardin/Drnevich constant.

The model adopted in FLAC2D generates backbone curves represented by Sun et al. (1998) for clay using gref 5 0:234 (Fig. 8) as the numerical fitting parameter. Later, similar to Case 2, the results of the fully nonlinear dynamic analyses in terms of base shears, lateral deflections, and interstory drifts under the influence of the earthquake records (Table 1) were determined from FLAC2D history records for flexible-base models resting on soil classes De and Ee .

Results and Discussion Because of word limitations of the paper, only full results of the lateral deflections for Model S15 resting on soil class Ee are presented in Fig. 9, as an example. To have a comprehensive comparison between the results of the equivalent linear and fully nonlinear dynamic analyses and to reach a clear conclusion about their accuracy in computing the inelastic seismic response of midrise building frames resting on soft soil deposits, the average values of the inelastic base shears (Table 4), lateral deflections, and interstory drifts (Figs. 10–15) for Models S5, S10, and S15 resting on soil

Fig. 11. Average seismic response of Model S5 resting on soil class Ee for three different cases: (a) lateral deflections; (b) interstory drifts

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classes De and Ee under the influence of the four mentioned earthquake records (Table 1) were determined and compared. In comparing the average base shear values of fixed-base and flexible-base models resting on soil classes De and Ee in Table 4, the average base shears, computed by the equivalent linear method, have reduced by 16, 20, and 32% in comparison with the fixed-base S5, S10, and S15 models, respectively. The average base shears of the flexible-base models, determined by the fully nonlinear method, have decreased by 20, 27, and 40% in comparison with the fixed-base models for Models S5, S10, and S15, respectively. Therefore, the base shears determined by the equivalent linear method are 4–8% more than the results, which are computed by the fully nonlinear method for seismic design of midrise building frames resting on soil class De . For the midrise building frames resting on soil class Ee , the average base shears, predicted by the equivalent linear method, have reduced by 24, 38, and 43% in comparison with the fixed-base S5, S10, and S15 models, respectively. The average base shears of the flexible-base models, determined by the fully nonlinear method, have reduced by 34, 50, and 54% in comparison with the fixed-base models for Models S5,

S10, and S15, respectively. Therefore, the base shears, computed by the equivalent linear method, are 10–13% more than the base shears, predicted by the fully nonlinear method for seismic design of midrise building frames resting on soil class Ee . Evidently, the differences between the base shear numerical predictions of the two methods in the case of the studied midrise building frames resting on soil classes De and Ee are insignificant. Therefore, the equivalent linear method may fairly predict the base shears and, subsequently, the internal forces of midrise building frames resting on soil classes De and Ee considering the SSI. However, comparing the average values of lateral deflections and interstory drifts of fixed-base and flexible-base models resting on soil classes De (Figs. 10, 12, and 14), it becomes apparent that the lateral deflections and corresponding interstore drifts of flexiblebase models, predicted by the equivalent linear method, have increased by 2, 5.5, and 11% in comparison with fixed-base S5, S10, and S15 models, respectively. However, the lateral deflections and interstory drifts of flexible-base models, determined by the fully nonlinear method, have amplified by 4, 11, and 20% in comparison with fixed-base models for Models S5, S10, and S15, respectively.

Fig. 12. Average seismic response of Model S10 resting on soil class De for three different cases: (a) lateral deflections; (b) interstory drifts

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Fig. 13. Average seismic response of Model S10 resting on soil class Ee for three different cases: (a) lateral deflections; (b) interstory drifts

By using the equivalent linear method, only half of the realistic seismic response, determined by the fully nonlinear method, can be achieved for seismic design of midrise building frames resting on soil class De . For the models resting on soil class Ee (Figs. 11, 13, and 15), lateral deflections and interstory drifts of flexible-base models, predicted by the equivalent linear method, have increased by 4, 10, and 23% in comparison with fixed-base S5, S10, and S15 models, respectively. However, the lateral deflections and interstory drifts of flexible-base models, determined by the fully nonlinear method, have enlarged by 11.5, 31, and 68%, comparatively. Consequently, the lateral deflection amplification caused by the SSI obtained through the equivalent linear method is almost one-third of the realistic deflection amplification determined by the fully nonlinear method for the midrise building frames resting on soil class Ee . As previously explained, the equivalent linear method does not directly capture any nonlinearity effects because of its linear solution process and takes the strain-dependent modulus and damping functions into account only in an average sense to approximate some effects of nonlinearity. However, the fully nonlinear method correctly represents the physics associated with the problem and follows more realistic stress-strain relationships, which enable the small strain shear modulus and damping degradation of the soil with © ASCE

strain level to be captured in the modeling. Based on the determined results, the linear solution process and the approximations that are used in the equivalent linear method result in significant inaccuracies in lateral deflection prediction of midrise building frames resting on soft soil deposits. This lack of accuracy may potentially underestimate the performance level of the building frames. For instance, based on the interstory results of Models S10 and S15 resting on soil class Ee [Figs. 13(b) and 15(b)], predicted by the equivalent linear method, the performance level of the models remain in the life safe level. However, realistic predictions (fully nonlinear method) suggest that the performance level of the models have been shifted from life safe to near collapse level as a result of soil-structure interaction effects. Such a considerable change in the performance level of the models can be dangerous and safety threatening. The equivalent linear method for dynamic analysis may underestimate the lateral deflections and corresponding interstory drifts of midrise building frames in comparison with the fully nonlinear dynamic analysis. Therefore, the dangerous and safety threatening effects of soil-structure interaction could be overlooked if the equivalent linear method is adopted in the seismic design procedure. Evidently, to guarantee the safety and integrity of the

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Fig. 14. Average seismic response of Model S15 resting on soil class De for three different cases: (a) lateral deflections; (b) interstory drifts

seismic design of midrise building frames taking into account the soil-structure interaction, it is recommended that the fully nonlinear method for dynamic analysis be used.

Conclusions In this study, to determine the accuracy and reliability of the equivalent linear analysis approach versus the fully nonlinear analysis method for seismic soil-structure interaction, numerical investigations using FLAC2D have been conducted. According to the numerical results, the discrepancies between the lateral deflections and interstory drifts, predicted by the equivalent linear method and the fully nonlinear method, can be up to 50 and 67% for the midrise building frames resting on soil classes De and Ee , respectively. Therefore, the equivalent linear method could, respectively, predict one-half and one-third of the realistic lateral deflections of midrise building frames resting on soil classes De and Ee . However, the discrepancies between the base shear results, © ASCE

determined by the equivalent linear method and the fully nonlinear method, are generally insignificant. The equivalent linear method can be used for computing the base shear and the subsequent internal forces of midrise building frames resting on soil classes De and Ee with acceptable accuracy. However, because of the significant differences between the lateral deflection predictions of the two methods in the cases studied, it is understood that the equivalent linear method of dynamic analysis may potentially underestimate the performance level of the mentioned building frames in comparison with the fully nonlinear dynamic analysis method. Therefore, the dangerous and safety threatening effects of soil-structure interaction could be overlooked and misinterpreted by using the equivalent linear method in the seismic design of midrise building frames resting on soft soils. Nevertheless, in some cases (e.g., Model S5 resting on soil class De ), this simplified approach may result in acceptable predictions. Although adopting performancebased design, the equivalent linear method of dynamic analysis may not be an accurate and qualified method for the seismic design and

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Fig. 15. Average seismic response of Model S15 resting on soil class Ee for three different cases: (a) lateral deflections; (b) interstory drifts

cannot adequately guarantee the structural safety of the midrise building frames resting on soft soil deposits.

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