Modelling Aspects In Ssi

ARTICLE IN PRESS

Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]] www.elsevier.com/locate/soildyn

Considering dynamic soil structure interaction (SSI) effects on seismic isolation retrofit efficiency and the importance of natural frequency ratio Edward H. Stehmeyer (III)a, Dimitris C. Rizosb, a

Structural Engineer, Collins Engineers, Inc., Charleston, S. Carolina, USA Department of Civil and Environmental Engineering, 300 Main, University of South Carolina, Columbia, SC 29208, USA

b

Received 27 November 2006; received in revised form 13 July 2007; accepted 19 July 2007

Abstract This paper utilizes and expands on existing coupled BEM–FEM (finite element method) methods for the investigation of the effects of soil structure interaction (SSI) on both an un-retrofitted and seismically isolated typical bridge structure. A simple numerical model of the bridge and surrounding soil is formulated and excited by an earthquake excitation. Utilizing Newmark’s b FEM solution method along with the closed form B-spline BIRF method, the structural damped period, composite damping ratio, pier relative displacement, and base shear demand are monitored. From these results, the effects of SSI on this structure are identified. Additionally, the importance of the relative rigidity between the soil-foundation system and the bridge structure is also investigated. The results of the studies indicate that the response of the complete structure system considered is affected by the inclusion of SSI effects. Furthermore, the efficiency of the isolation measures designed using fixed base conditions is decreased by considering SSI over a certain relative rigidity range that is quantified using the structure to soil-foundation natural frequency ratio. r 2007 Elsevier Ltd. All rights reserved. Keywords: Dynamic soil structure interaction; Seismic isolation; Isolator efficiency; Bridge seismic retrofit; Natural frequency ratio

1. Introduction Along with the evolution of construction practices and design standards for civil infrastructures, there is a growing demand for assessment and identification of structural components that may require replacement or retrofit. One aspect of design that has necessitated this inquiry is the introduction of stricter seismic design criteria and advanced analysis methods for buildings and bridges. Consequently, the capacity and performance of existing bridges and buildings that make up the current infrastructure may not meet the new design criteria. This realization has spurred a number of seismic evaluation programs that address current risk to infrastructure elements and identify potential structures for retrofit.

Corresponding author. Tel.: +1 803 777 6166; fax: +1 803 777 0670.

E-mail address: [email protected] (D.C. Rizos).

When a structure is deemed deficient by current seismic standards, there are three alternatives for updating the structure. The inadequate components can either be completely replaced, strengthened in both capacity and ductility, or retrofitted so that the structure can withstand the appropriate earthquake design demand. Frequently, upgrading the deficient structure with new retrofit technology is more economical than complete replacement or structural component improvement [1]. A popular method of seismic retrofit is the use of seismic isolation devices placed between the superstructure (bridge deck or above stories) and the substructure (columns/bents or foundation). Such devices uncouple the motion of critical structural components and, the seismic demands on the existing structure are reduced through the isolator’s natural action of period elongation, increased damping, and energy dissipation [2]. It should be noted that the natural period of the isolated structure tends to be much longer than the predominant periods of imposed ground motion [3,4].

0267-7261/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2007.07.008 Please cite this article as: Stehmeyer EH, Rizos DC. Considering dynamic soil structure interaction (SSI) effects on seismic isolation retrofit efficiency and the importance of natural frequency.... Soil Dyn Earthquake Eng (2007), doi:10.1016/j.soildyn.2007.07.008

ARTICLE IN PRESS 2

E.H. Stehmeyer, D.C. Rizos / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]

Often, seismic structural design is based on the rigid base assumption, and interaction with the soil-foundation system is either ignored or carried out separately, when in reality these systems are coupled. Ignoring the SSI effects may lead to erroneous structural assessment and estimates of seismic demands. Although SSI effects may not be important for all structures, it has been recognized that they may have a significant impact on the dynamic system response, especially in cases involving heavier structures and soft soil conditions. The impact of considering SSI effects has been reported in the literature as, natural period elongation [5,6] and added composite damping [6,7]. For structures founded on soft soils with high relative rigidity with respect to the supporting soil, including SSI effects amplifies the dynamic response of the system [7,8]. Considering SSI effects in an analysis of structures with seismic isolation devices, or other period lengthening devices such as tuned mass dampers, has been shown to decrease the effectiveness of the retrofit measure [9]. The difficulty in modeling the complicated soil-foundation-structure system has been the topic of many research efforts since the mid-1960s. Methods for seismic and dynamic analysis of soils and structures, including SSI effects, are based on analytical, experimental and numerical procedures in conjunction with observations of physical behavior and lessons learned from past events. The finite element method (FEM), the boundary element method (BEM), and coupled BEM–FEM are among the most popular numerical techniques for rigorous modeling of a soil-foundation-structure system. Literature reviews on BEM and FEM methods with applications in SSI analysis have been presented by Beskos [10], among others. Such methods yield accurate results but tend to be computationally expensive. Due to this expense further simplifying assumptions for the modeling of the soilfoundation-structure system have led to lumped parameter models and closed form solutions that, while approximate in nature, have been shown to generate similar accuracy as rigorous methods [11–13]. The use of rigorous and simplified numerical models to represent the soil-foundation portion of a complete structural system has been shown to be accurate, and particularly attractive for use in BEM–FEM coupled modeling of complete infrastructure systems including bridges and high speed rail systems [11,12,14,15]. This work presents a new, simplified, yet accurate, procedure for the rapid assessment of the effectiveness of seismic isolation devices when SSI effects are accounted for. Furthermore, it investigates the importance of SSI phenomena on the response of seismically isolated bridges. To this end, the closed form B-spline impulse response function (BIRF) method introduced by the authors [12] is coupled with the Newmark’s b FEM procedure to model a typical bridge structure within the general framework of the staggered BEM–FEM approach introduced in [14]. The bridge structure is analyzed under seismic loading and the

effects of SSI are identified. The bridge is then considered for a seismic isolation retrofit based on a typical fixed base analysis design procedure [2]. The inclusion of SSI in the analysis of the retrofitted structure is then used to evaluate the efficiency of the isolation system. Finally, a parametric study is conducted to investigate the relationship of relative rigidity in the complete structural system by studying the importance of structure to soil natural frequency ratio through displacement and base shear force demand comparisons. 2. Numerical solution procedures In this work, the bridge-foundation-soil system is modeled using a staggered coupled BEM–FEM approach as first introduced in [14] and modified herein. In particular, the bridge and the isolation system, when present, are modeled using the Newmark’s b FEM method while the soil-foundation system is modeled by implementing the closed form B-spline BIRF solution method [12]. A brief overview of each solution method is presented next followed by the adaptation of the staggered coupled BEM–FEM approach for use with the closed form B-spline BIRF solution method. 2.1. FEM and Newmark’s b direct time integration The FEM method is used in this study for modeling the bridge structure system. The governing equation of motion for a multi-degree of freedom (MDOF) system can be represented in the following semi-discrete form M€u þ C_u þ Ku ¼ P,

(1)

where M, C, and K are the mass, damping, and stiffness coefficient matrices that are developed based on standard FEM procedures, u is the displacement vector, and dots represent derivatives with respect to time. The load vector P represents all of the transient forces acting on, or applied to, the system. Eq. (1) can be partitioned with respect to the degrees of freedom that are free to move (f) and those that are in contact (c) with the soil-foundation system as " #( ) " #( ) Mff Mfc Cff Cfc u€ f u_ f þ Mcf Mcc Ccf Ccc u€ c u_ c " #( ) ( ) uf Kff Kfc Pf þ ¼ . ð2Þ Kcf Kcc uc Pc The load vector P can also be expanded further to separate the contributing excitation components as ( ) ( n ) Pf þ Peq Pf f P¼ ¼ , (3) Rc þ Peq Pc c where the load vector contains nodal forces, Pnf applied directly to the free nodes of an element, and nodal equivalent forces, Peq f , that account for body forces, and other forces not applied directly to the free nodes. The

Please cite this article as: Stehmeyer EH, Rizos DC. Considering dynamic soil structure interaction (SSI) effects on seismic isolation retrofit efficiency and the importance of natural frequency.... Soil Dyn Earthquake Eng (2007), doi:10.1016/j.soildyn.2007.07.008

ARTICLE IN PRESS E.H. Stehmeyer, D.C. Rizos / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]

contact forces include support reactions, Rc , as well as nodal equivalent forces Peq c . For direct time domain solutions, the governing equation, Eq. (1), is integrated directly in time by employing Newmark’s b method. Applying Newmark’s b method, with a time step dt, the governing equation of motion can be cast in a form of system of algebraic equations that can be solved in an explicit time marching scheme for the forward step N þ 1 as [16] " #( )Nþ1 ( )Nþ1 ( eq )Nþ1 ^f Dff Dfc uf P^ f P ¼ þ , (4) Dcf Dcc uc Peq Rc c ^ are, respectively, defined as where D and P D¼Kþ Nþ1

^ P

1 d C, Mþ 2 bdt bdt

(5)

   1 N 1 N 1 _  1 u€ N ¼P þM u þ þ u bdt2 bdt 2b       g N g dt g u þ  1 u_ N þ  2 u€ N , ð6Þ þC bdt b 2 b 

Nþ1

where b and g are constants associated with Newmark’s b method. In order to solve this system of equations, we assume that the displacement of the contact (c) degrees of freedom, uc , is known at the forward step N þ 1 either explicitly or as computed or predicted by the BEM method, as in the next section. Consequently, the only unknowns in Eq. (4) are the displacement at the free degrees of freedom, uf and the reaction forces, Rc which are computed in a single step as ( )Nþ1 " # 8( )Nþ1 8 eq 9Nþ1 < P^ f = ^f uf Dff 0 1 < P ¼ þ : 0 : Peq ; Dcf I Rc c " 

Dfc Dcc

#

9 =

fuc gNþ1 , ;

ð7Þ

where I is the identity matrix. 2.2. Closed form B-spline BIRF solution method The supporting soil-foundation system is modeled using the closed form B-spline impulse response function (BIRF) method for rigid surface square foundations resting on homogeneous elastic soils, as reported by the authors [12]. The BIRF represents the time history of the displacement of the soil-foundation system when the foundation is excited by transient impulse forces and moments of B-spline polynomial modulation and duration Dt. Due to symmetries, four primary modes of vibration are considered, i.e., the horizontal, vertical, rocking, and torsion, while the coupling modes are negligible for this class of problems and ignored in this study. The corresponding BIRFs for the horizontal, HðtÞ, vertical, V ðtÞ, rocking RðtÞ,

3

and torsion, TðtÞ modes are given in a closed form as vs Dt vs Dt HðtÞ; V z ðtÞ ¼ V ðtÞ, 2 Gw Gw2 vs Dt vs Dt Rx ðtÞ ¼ Ry ðtÞ ¼ RðtÞ; T z ðtÞ ¼ TðtÞ, ð8Þ Gw4 Gw4 where G and vs represent the shear modulus and shear wave velocity of the soil, w is the foundation width and t ¼ tvs =w is a non-dimensional time. The non-dimensional BIRFs are given in terms of the B-spline support (impulse duration), Dt, as 8 0:2011 > > 0ptpDt; > < Dt t; HðtÞ ¼ ½ð0:23Þ sinhð9:47ðt  DtÞÞ þ 0:2011 > > > :  coshð9:47ðt  DtÞÞe12:23ðtDtÞ ; Dtot;

H x ðtÞ ¼ H y ðtÞ ¼

8 0:1051 > > t; > < Dt V ðtÞ ¼ ½ð0:239Þ sinhð1:6ðt  DtÞÞ þ 0:1051 > > > :  coshð1:6ðt  DtÞÞe3:85ðtDtÞ ;

0ptpDt; Dtot;

8 1:082 > > 0ptpDt; > < Dt t; RðtÞ ¼ ½1:082 cosð3:18ðt  DtÞÞ þ ð0:365Þ > > > :  sinð3:18ðt  DtÞÞe2:61ðtDtÞ ; Dtot; 8 1:016 > > > < Dt t; TðtÞ ¼ ½1:016 cosð2:47ðt  DtÞÞ þ ð0:269Þ > > > :  sinð2:47ðt  DtÞÞe2:89ðtDtÞ ;

0ptpDt; Dtot: (9)

These closed form solutions have demonstrated equivalent accuracy and versatility for direct time domain modeling of dynamic SSI problems with rigid surface foundations as compared to more rigorous boundary element and finite element methods. Computational efficiency is gained through the implementation of these closed form solutions in place of the rigorous evaluation of the BIRF for different foundation sizes and soil types. Each of the BIRFs can be thought of as the response of a lumped parameter model associated to a vibration mode of the soilfoundation system. The computational effort is similar to the implementation of lumped parameter models. The closed form BIRF solution method has several advantages over the usual lumped parameter models. They have shown better accuracy than frequency independent lumped parameter models, and are less complex than frequency dependent models. The closed form solutions are also appropriate for direct time domain analysis and therefore suitable for non-linear analysis of the structure. The method accommodates any time step suitable for the specific soil type considered and has no limitation on the type, duration, and frequency of loading since the BIRF

Please cite this article as: Stehmeyer EH, Rizos DC. Considering dynamic soil structure interaction (SSI) effects on seismic isolation retrofit efficiency and the importance of natural frequency.... Soil Dyn Earthquake Eng (2007), doi:10.1016/j.soildyn.2007.07.008

ARTICLE IN PRESS E.H. Stehmeyer, D.C. Rizos / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]

4

captures the dependency on frequency implicitly. Further details and validation studies for these closed form BIRF solutions have been reported by the authors in [12]. These closed form BIRF solutions are used within the B-spline impulse response technique as reported in [14] to obtain the three translations and three rotations, uRF , of the Rigid Foundation system to arbitrary loading, PRF ðtÞ, as uN RF ¼

N þ1 X

ðBIRFÞn PNnþ2 , RF

(10)

n¼1

where BIRF represents any of the six responses indicated in Eq. (8) and superscripts indicate the time step at which quantities are evaluated. This expression represents the response of the six degrees of freedom in contact with the structure and is coupled to the response of the structure, Eq. (7), as discussed in the next section. While the response of the free field of the soil-foundation system is not explicitly calculated here, its influence on the soil-foundation system response is inherently accounted for in the closed form BIRF solutions. 2.3. Staggered coupled closed form BIRF–FEM method Since the soil-foundation-structure system is physically coupled, their governing equations cannot be solved independently of one another. The coupled BEM–FEM staggered solution approach developed by Rizos et al. [14] is adapted herein by incorporating the closed form BIRF solutions in place of the B-spline BIRF BEM. In following with the original scheme, the structure is modeled by the FEM method, the soil-foundation system is modeled by the closed form BIRF solutions and the coupling of the two domains is performed at the contact nodes (c) on the structure-foundation level by enforcing equilibrium and compatibility conditions in a staggered time marching scheme as shown in Fig. 1(a). The solution process begins with the FEM solver (Eq. (7)) at point A shown in Fig. 1(a). Any combination of prescribed structure initial displacements, initial velocities, nodal forces, nodal equivalent forces, or prescribed support displacements are applied on the structure. The contact node values are assumed known at this step and are kept fixed in space during the current time step. The FEM solver, (Eq. (7)), is activated to compute the solution at time ðN þ 1Þdt, depicted in Fig. 1(a) as point B. The FEM solver then provides the contact node resultant force to the BIRF Solver (Eq. (10)) by enforcing equilibrium, at the contact nodes, i.e., Rc ¼ PRF . This equilibrium establishes the influence of the structure on the soil-foundation system. The closed form BIRF Solver then computes the new nodal values (e.g. displacements) at the contact nodes, uRF (Eq. (10)), for time ðN þ 1Þdt, in response to the structure behavior, depicted as point D in Fig. 1(a). This completes the calculations at the current time step N þ 1. Subsequently, the closed form BIRF solver, through compat-

Fig. 1. Coupled closed form BIRF–FEM: (a) staggered time solution scheme and (b) solution flowchart.

ibility of displacements, uc ¼ uRF provides the contact nodal values for use by the FEM solver at the forward time ðN þ 2Þdt, depicted as point E in Fig. 1(a). It should be noted that all inertia contributions, such as structural and foundation mass, are accounted for in the FEM model. A computation flowchart is provided in Fig. 1(b). 3. Modeling aspects The modeling details of the bridge pier and deck, seismic isolators, soil-foundation system and the time history analysis method used in this work are discussed next. The physical model represents the transverse cross-section of a typical bridge structure as shown in Fig. 2(a). The structure system consists of three major components, i.e., the superstructure, the pier, and the seismic isolation device(s). The following sections introduce the details of each component and discuss the development of the associated numerical models that are suitable for the proposed SSI analysis methods. 3.1. Description of bridge structure The bridge has a reinforced concrete pier with a circular cross-section of diameter d ¼ 5 ft (1.524 m). The gross section properties are used in this example and cracking is not considered. The height of the column is L ¼ 30 ft (9.144 m). A cross-section of the bridge deck is assumed from which the tributary lumped mass over the pier is assumed to be M deck ¼ 2614:13 lb s2 =in ð46; 704:8 kg s2 =mÞ as shown in Fig. 2(b). Also shown in Fig. 2(b), the bridge pier is modeled using a 2 node 3-D prismatic beam finite element. The modulus of elasticity is assumed to be E c ¼ 3122018:6 psi (21525.56 MPa) for normal weight concrete and the Poisson’s ratio is n ¼ 0:15. The cross-sectional area

Please cite this article as: Stehmeyer EH, Rizos DC. Considering dynamic soil structure interaction (SSI) effects on seismic isolation retrofit efficiency and the importance of natural frequency.... Soil Dyn Earthquake Eng (2007), doi:10.1016/j.soildyn.2007.07.008

ARTICLE IN PRESS E.H. Stehmeyer, D.C. Rizos / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]

Fig. 2. (a) Physical model of the transverse cross-section of a typical bridge structure and (b) soil-foundation-structure numerical model including a lumped parameter equivalent linear isolator numerical model.

is A ¼ 2827:4 in2 ð1:824 m2 Þ, the moment of inertia is I ¼ 6:36  105 in4 ð0:265 m4 Þ, and the polar moment of Inertia is J ¼ 12:72  105 in4 ð0:529 m4 Þ. The mass matrix, M is assumed diagonal and considers inertia effects in the translational degrees of freedom only. The damping coefficient matrix, C, is determined based on Rayleigh damping as C ¼ a0 M þ a1 K,

(11)

with a critical damping ratio, x of 5% and a0 ¼ 0:9267 and a1 ¼ 0:0024 based on the target frequencies of 1:0ons and 2:0ons where ons ¼ 6:95 cycles=s is the natural frequency of the structure assuming rigid base conditions [17]. 3.2. Seismic isolator models Seismic isolators have been modeled in a variety of different ways including early representations as soft stories for base isolation of buildings [18], simple shear springs [3], inelastic truss elements [19], and have even been developed for use in complex finite element procedures [20,21]. Often, the non-linear nature of an isolator is

5

modeled as an ‘‘equivalent’’ linear system with optimal stiffness and damping that is indicative of the average stiffness variability and dissipation characteristics of the bearing. In particular, typical modeling of isolators is usually represented by either an equivalent linear or bilinear model as shown in Fig. 2(b) to account for the load displacement behavior of various types of isolation bearings [22,23]. This type of model has enabled several research efforts on the effectiveness of bridge isolation retrofits with bi-directional excitations [22] and the effects of pier and deck flexibility on the response of isolated bridges [23]. In this work an equivalent linear stiffness and damping model is utilized to represent a high damping laminated rubber (HDLR) isolator with stiffness, K iso , and equivalent damping ratio, xiso as shown in Fig. 2(b). The equivalent damping ratio, xiso is the area formed by the hysteresis loop of the load deformation behavior of the isolator, or can be determined via manufacturer testing [2]. The seismic isolation device implemented in this study is a 3-D linearized model consisting of two nodes connecting the top of the pier to the mass of the deck with only translational degrees of freedom at each end, and lumped mass and damping as illustrated in Fig. 2(b). The transverse direction horizontal natural period, T n and frequency, ons of the un-isolated structure are T n ¼ 0:9 s and ons ¼ 6:95 cycles=s, respectively. Two isolators, ‘‘Iso. Dev. #1’’ and ‘‘Iso. Dev. #2’’ are selected so that values of 2.0 and 3.0 s for the target natural period of vibration, T n target , for the isolated structure are obtained. The effective horizontal stiffness of the isolated structure is calculated as [2]  2 2p K eff ¼ M deck . (12) T n target As can be seen from Fig. 2(b), the isolator and pier models are connected in series, therefore the effective stiffness of the isolation system, K iso , can be calculated as K iso ¼

K eff K beam . ðK beam  K eff Þ

(13)

For T n target ¼ 2:0 s, the total bearing horizontal effective stiffness is K Iso Dev 1 ¼ 32; 332:3 lb=in ð5662:25 k N=mÞ, and for T n target ¼ 3:0 s, K Iso Dev 2 ¼ 12; 598 lb=in ð2206:25 k N=mÞ. Typical values of elastomeric isolator horizontal stiffness for bridge structures are on the order of 5833.33–11,667.67 lb/in (1021.57–2043.32 k N/m) [2]. Therefore, for the target period of 2.0 s (Iso. Dev. #1), the resulting isolator is slightly stiffer than the typical range. For the target period of 3.0 s, Iso. Dev. #2 is on the softer end of the range of typical values for a 2 bearing retrofit. This analysis is conducted to provide insight into the effectiveness of such retrofit measures when including SSI effects. Note that the value of K Iso Dev# used in the following analysis for bearing horizontal stiffness is the lumped or total stiffness of the proposed isolators. This value should be divided by the number of isolators to obtain a discrete value of horizontal stiffness for the design of each bearing.

Please cite this article as: Stehmeyer EH, Rizos DC. Considering dynamic soil structure interaction (SSI) effects on seismic isolation retrofit efficiency and the importance of natural frequency.... Soil Dyn Earthquake Eng (2007), doi:10.1016/j.soildyn.2007.07.008

ARTICLE IN PRESS 6

E.H. Stehmeyer, D.C. Rizos / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]

The total horizontal damping coefficient for the isolator, ciso , can then be computed for an assumed value of the equivalent damping ratio for the isolator as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (14) ciso ¼ xiso 2 K iso  M deck . In this work, it has been assumed that xiso ¼ 15%, which is representative of an HDLR isolator [2]. The deck mass, M deck , is used in the calculation since the isolation system de-couples it from the bridge pier. Once all quantities are known, simple superposition of the matrices for stiffness, mass, and damping of the isolator into the global structural matrix for the isolated structure is preformed. 3.3. Soil-foundation closed form BIRF simplified models The closed form BIRF models used within this work represent the soil-foundation system response as an equivalent simplified numerical model for each soilfoundation vibration mode as shown in Fig. 3(a). While

this representation is an approximation, it has been shown to produce accurate soil-foundation system responses in the time and frequency domain [11,12]. In this study, the closed form BIRF solutions presented in Eqs. (8) and (9) are utilized. The properties of the non-dimensional equivalent SDOF systems for a rigid soil-foundation system are listed in Table 1 [12]. In particular, the non-dimensional natural frequency, on RF and damping ratio, xRF of the rigid square surface foundation system are presented for the horizontal, HðtÞ, vertical, V ðtÞ, rocking, RðtÞ, and torsion, TðtÞ modes of vibration. Using a consistent de-normalization method as was used for the closed form BIRF solutions, it can be shown that, for a particular soil with shear modulus, G s , and density, r, and a square foundation of width w, the equivalent dimensional natural frequency of the rigid soil-foundation system is calculated as on RF

sffiffiffiffiffiffiffiffi Gs . ¼ on RF rw2

(15)

Fig. 3. (a) Simplified lumped parameter numerical model incorporating the closed form BIRF solutions and equivalent linear isolator model and (b) the measured relative pier displacement in the numerical model. Please cite this article as: Stehmeyer EH, Rizos DC. Considering dynamic soil structure interaction (SSI) effects on seismic isolation retrofit efficiency and the importance of natural frequency.... Soil Dyn Earthquake Eng (2007), doi:10.1016/j.soildyn.2007.07.008

ARTICLE IN PRESS E.H. Stehmeyer, D.C. Rizos / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]

7

Table 1 Non-dimensional equivalent SDOF properties for the soil-foundation system [12] Parameter

Mode

Natural frequency, on RF Damping ratio, xRF

Horizontal H

Vertical V

Rocking R

Twist T

7.74 1.58

3.5 1.1

4.1138 0.6343

3.8 0.76

Table 2 Soils considered and their respective properties Soila #

1 2 3 4 5 6 7 8

Density r ðSlugs=in3 Þ ðkg=m3 Þ 0.000153 0.000142 0.000167 0.000172 0.000177 0.000181 0.000192 0.000202

a

(0.097) (0.0897) (0.106) (0.108) (0.111) (0.114) (0.121) (0.127)

Soil shear modulus G s (psi) (MPa)

750.02 1687.54 2000.00 2437.56 4500.00 6562.66 8900.00 11250.28

(5.17) (11.64) (13.79) (16.81) (31.03) (45.25) (61.36) (77.57)

Soil shear wave velocity vs (in/s) (cm/s)

2214.07 3447.33 3460.64 3764.55 5042.19 6021.44 6808.39 7462.87

(5623.75) (8756.24) (8790.04) (9561.98) (12807.19) (15294.49) (17293.35) (18955.73)

Horizontal model equivalent soil natural frequency on soil (cycles/s) 142.81 222.35 223.21 242.81 325.22 388.38 439.14 481.36

Poisson’s ratio ¼ 13.

It should be noted that in this work, that since the structure is not changed, the foundation side width, w is also kept constant and w ¼ 120 in (3.048 m). Eight soils are considered for use in the parametric SSI studies of this work that range from soft (soil 1) to stiff media (soil 8) with properties and calculated equivalent horizontal dimensional natural frequencies listed in Table 2. 3.4. Time history analysis The complete soil-foundation-bridge model is subjected to seismic loading time history records from the Imperial Valley, El Centro 1940 E-W component. Through time history analysis, the impact of SSI on the structural response for isolated and non-isolated conditions is then quantified using rigid base analysis of the structure as a baseline. To this end, the acceleration time history vector for the Imperial Valley, El Centro 1940 E-W component, aðtÞ was used along with the superstructure lumped mass, M deck , to obtain an equivalent force time history vector PðtÞ as PðtÞ ¼ M deck aðtÞ.

(16)

This force time history was applied to the lumped mass at the superstructure level for all comparative studies as shown in Fig. 3(a). During these studies, the maximum relative top of pier displacement, D, as shown measured in Fig. 3(b), base shear responses, composite damping ratio, xn , and damped period of vibration, T D , were monitored. The composite damping ratio of the system was calculated via a logarithmic decrement formulation under the assumption of a small damping ratio with measurements

taken in the free vibration phase of the system response and reflects the coupled behavior of the soil-foundationstructure system [17]. 4. Analysis results and discussion This section presents the analysis results and discusses the importance of SSI effects on the efficiency of seismic isolation systems. To this end, the concept of the relative natural frequency ratio between the structure and the soilfoundation system is introduced. 4.1. Effects of SSI on the structure without seismic isolation A time history analysis was performed using a rigid base assumption and the proposed coupled closed form BIRF–FEM method to account for SSI effects. Figs. 4(a)–(e) show the time history of the horizontal displacement of the top of pier relative to the foundation for a structure without seismic isolation. Five analysis cases including rigid base analysis, and SSI analysis using soils 6, 4, 2 and 1 are considered, respectively. It is observed that the SSI effects are more pronounced for softer soils (soil 1) and less significant for stiffer soils (soil 6). In particular, the effects of SSI on this structure are: (i) increased maximum relative pier displacement, (ii) increased number of significant cycles of large amplitude displacement, and (iii) significant lengthening of the damped structural period of vibration for softer soils. In order to quantify the lengthening of the damped structural period for un-isolated and isolated structures, the normalized damped period ratio ðT D =T DRB Þ is plotted

Please cite this article as: Stehmeyer EH, Rizos DC. Considering dynamic soil structure interaction (SSI) effects on seismic isolation retrofit efficiency and the importance of natural frequency.... Soil Dyn Earthquake Eng (2007), doi:10.1016/j.soildyn.2007.07.008

ARTICLE IN PRESS E.H. Stehmeyer, D.C. Rizos / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]

8

f 6.0 5.0 4.0 3.0 2.0 1.0 0.0 -1.0 -2.0 -3.0 -4.0 -5.0

Rigid Base Analysis

Displacement (in)

Displacement (in)

a

SSI - Soil 6

SSI - Soil 4

d

Iso. Dev. #2 SSI - Soil 6

3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5

Iso. Dev. #2 SSI - Soil 4

i 6.0 5.0 4.0 3.0 2.0 1.0 0.0 -1.0 -2.0 -3.0 -4.0 -5.0

SSI - Soil 2 Displacement (in)

Displacement (in)

3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5

h 6.0 5.0 4.0 3.0 2.0 1.0 0.0 -1.0 -2.0 -3.0 -4.0 -5.0

Displacement (in)

Displacement (in)

c

e

3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5

Iso. Dev. #2 SSI - Soil 2

j 6.0 5.0 4.0 3.0 2.0 1.0 0.0 -1.0 -2.0 -3.0 -4.0 -5.0

SSI - Soil 1 Displacement (in)

Displacement (in)

Iso. Dev. #2 Rigid Base Analysis

g 6.0 5.0 4.0 3.0 2.0 1.0 0.0 -1.0 -2.0 -3.0 -4.0 -5.0

Displacement (in)

Displacement (in)

b

3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5

0

5

10

15

20

25

30

Time (sec)

35

40

45

50

55

3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5

Iso. Dev. #2 SSI - Soil 1

0

5

10 15 20 25 30 35 40 45 50 55 60 65 Time (sec)

Fig. 4. Top of the pier relative horizontal response to El Centro 1940 for rigid base and varying soil conditions: (a)–(e) no seismic isolation and (f)–(j) Isolation Device #2.

ARTICLE IN PRESS E.H. Stehmeyer, D.C. Rizos / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]

versus the soil shear modulus, Gs , for the cases of rigid base ðT DRB Þ and SSI ðT D Þ analysis in Fig. 5(a). This plot shows the large lengthening of the period of the structure for ‘‘softer’’ soils for both types of structures. The trend, as mentioned earlier, is to lengthen the period for less stiff soils and to converge to the natural period of the structure under rigid base analysis for stiffer soils. The normalized composite damping ratio, xn =xn RB , is also plotted in Fig. 5(a) for the SSI analysis and illustrates the higher damping ratio for softer soils that eventually approaches the rigid base damping ratio for stiffer soils in un-isolated structures. This higher value of composite damping ratio is attributed to the added radiation damping that is present when the stiffness of the soil is included in the analysis. It is

9

also interesting to note that the composite damping ratio of the isolated systems is lower than the rigid base analysis case indicating some counteraction of the composite damping for isolated structures on softer soils. 4.2. Efficiency of isolation retrofit considering SSI The isolated bridge shown in Fig. 3(a) is subjected to the same equivalent force time history from El Centro 1940 for two different seismic isolation retrofits (Iso. Dev. #1 & #2). Rigid base analysis of the seismically isolated structures is performed and the relative horizontal displacement of the top of the pier is monitored. The combined effects of considering SSI along with seismic isolation of the two

4.5

4.5

4

Iso. Dev. #1 - Normalized Period Ratio

4

3.5

Iso. Dev. #2 - Normalized Period Ratio

3.5

No Isolation - Normalized Composite Damping Ratio

3

3 Iso. Dev. #1- Normalized Composite Damping Ratio

2.5

Iso. Dev. #2 - Normalized Composite Damping Ratio

2

2.5 2

1.5

1.5

1

1

Rigid Base Analysis

0.5

0.5 0 0

2000

4000 6000 8000 Shear Modulus, G (psi)

10000

Normalized Composite Damping Ratio n /nRB

Normalized Damped Period Ratio TD / TDRB

No Isolation - Normalized Period Ratio

0 12000

Pier Relative Maximum Displacement (in)

7 6 5

Rigid Base

No Iso. Dev. Iso. Dev. #1 Iso. Dev. #2

SSI

No Iso. Dev. Iso. Dev. #1 Iso. Dev. #2

4 3 2 1 0 0

2000

4000 6000 8000 Shear Modulus, G (psi)

10000

12000

Fig. 5. Structural response SSI effects when subjected to El Centro 1940 including: (a) normalized damped period ratio and normalized composite damping ratio and (b) pier relative maximum displacement. Please cite this article as: Stehmeyer EH, Rizos DC. Considering dynamic soil structure interaction (SSI) effects on seismic isolation retrofit efficiency and the importance of natural frequency.... Soil Dyn Earthquake Eng (2007), doi:10.1016/j.soildyn.2007.07.008

ARTICLE IN PRESS 10

E.H. Stehmeyer, D.C. Rizos / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]

isolated bridge models is investigated next using the same excitation while monitoring the relative pier horizontal displacement. Figs. 4(f)–(j) show the results of the relative pier displacements retrofitted with Iso. Dev. #2 as compared to the rigid base analysis results for soils 6, 4, 2 and 1, respectively. SSI effects can be seen in the response amplification of the pier for the softer soils 1, 2 and 4. It is interesting to note that the pier relative displacement increases drastically compared to the displacements of rigid base analysis. When SSI is considered, a period shift is noticeable for each of the soils with the softest soil (soil 1) having the most pronounced elongation. Also, for these soils the composite damping ratio appears to be very small, suggesting that SSI effects counteract the damping characteristics of the isolation system. It should also be noted that as the soil gets stiffer (soil 6), the effects of SSI are less significant in the response of the isolator, but still tend to amplify the response of the pier by approximately 40%. Fig. 5(b) illustrates the effects of SSI on the isolated structures by graphing the top of pier relative maximum displacement versus the soil shear modulus, G s , for isolated and un-isolated cases utilizing rigid base and SSI analysis. While both isolators reduce the demand on the pier under rigid base conditions, the consideration of SSI for this particular isolation retrofit scheme lessens the effectiveness of the isolator to reduce the demands on the bridge pier for softer soils when a base isolation design is performed based on rigid base analysis. Similar results were obtained for the retrofit with Iso. Dev. #1 [11]. 4.3. Importance of relative natural frequency ratio Often SSI effects are identified as important for massive structures on soft soils. The bridge structure chosen in this study is relatively light when compared to a nuclear energy facility; however, SSI effects are important for a range of soil conditions. Therefore, focusing exclusively on a structure’s stiffness or mass appears to be insufficient to judge the importance of SSI on a structure. Rather, the assessment of the relative rigidity quantified through a natural frequency ratio of the structure to the soil is proposed in this work to generalize the importance of SSI effects and their effect on the effectiveness of isolation retrofit. The bridge shown in Fig. 3(a) is once again subjected to a sinusoidal load of the form PðtÞ ¼ P0 sinðOtÞ,

(17)

where P0 is the amplitude, and O is the frequency of the excitation. In this study, a 10,000 pound (4535.9 kg) amplitude sin wave with frequencies proportional to the un-isolated structure natural period, ons and isolated (Iso. Dev. #2) structure natural period, onsI are assumed. In particular, excitation frequencies of O ¼ 0:5ons , 1:0ons , 1:5ons , and 1:0onsI are applied to the numerical model, the response is calculated as discussed previously, and the top

of pier relative displacement and base shear maximum responses are monitored. The properties of the structure and foundation are kept constant and only the shear modulus of the soil, Gs , and density, r, varies. The natural frequency ratio, FR adopted in this study is calculated as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K H structure =M deck ons rffiffiffiffiffiffiffiffi FR ¼ ¼ , (18) on RF Gs on RF rw2 where K H structure is the horizontal stiffness of the pier represented by the 3-D FEM beam element. The values of FR used ranged from 0.007 to 0.168 where the lower ratio indicates a very stiff soil, whereas, the larger ratio represents a very soft soil condition. Fig. 6(a) shows the maximum horizontal displacements of the pier during the frequency ratio study for the retrofitted and non-isolated structures. As can be seen from this figure, the effects of SSI are present in the form of period elongation. Since the period is inversely proportional to the frequency of the structure, the application of a sinusoidal load below the resonant natural frequency of the structure considering rigid base conditions produces larger displacement demands on the pier. Both the application of 0:5ons and 1:0onsI cause this increase in maximum relative displacement since they are most likely closer to the natural frequency of the structure when SSI effects are taken into account. Another important observation from Fig. 6(a) is that the efficiency of Iso. Dev. #2 varies over the natural frequency ratio range studied. The isolation retrofit is only effective for the first portion of the frequency ratio range. The pier displacement demands are actually increased for the isolated structure above FR ¼ 0:05520:065 for all of the excitation frequencies investigated as indicated by the dotted circles in the figure. This figure also demonstrates that the inclusion of SSI along with isolation tends to reduce the effectiveness of the isolation scheme for moderate to soft soil conditions for this structure using the rigid base isolation retrofit design. Similarly, in Fig. 6(b) the base shear demand on the pier is illustrated over the same natural frequency ratio range for the various harmonic excitations. The isolation retrofit is effective in the lower natural frequency ratio range while becoming ineffective after FR ¼ 0:7520:9. Based on these results, it is recommended that an evaluation of the effects of SSI on seismically isolated structures should be performed and incorporated into the design process for seismic isolation retrofit of structures. While these effects may not be important for every structure, care should be exercised for structures above the natural frequency ratio ranges identified. 5. Conclusions A new simplified, yet accurate, procedure for the rapid assessment of the effectiveness of seismic isolation devices including SSI phenomena on the response of seismically

Please cite this article as: Stehmeyer EH, Rizos DC. Considering dynamic soil structure interaction (SSI) effects on seismic isolation retrofit efficiency and the importance of natural frequency.... Soil Dyn Earthquake Eng (2007), doi:10.1016/j.soildyn.2007.07.008

ARTICLE IN PRESS E.H. Stehmeyer, D.C. Rizos / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]

Pier Relative Maximum Horizontal Amplitude (in)

3.5

No Iso. Dev.

Iso. Dev. #2

 = 0.5 ns  = 1.0 ns  = 1.5 ns  = 1.0 ns-I

3.0

11

 = 0.5 ns  = 1.0 ns  = 1.5 ns  = 1.0 ns-I

2.5 2.0 1.5 1.0 0.5 0.0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Natural Frequency Ratio (Structure/Soil)

1400 No Iso. Dev.

Max Base Shear (lb)

Iso. Dev. #2

 = 0.5 ns  = 1.0 ns  = 1.5 ns  = 1.0 ns-I

1200 1000

 = 0.5 ns  = 1.0 ns  = 1.5 ns  = 1.0 ns-I

800 600 400 200 0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Natural Frequency Ratio (Structure/Soil) Fig. 6. Frequency ratio study: (a) pier relative maximum horizontal displacement and (b) maximum pier base shear for isolated and un-isolated cases by varying the soil shear modulus under harmonic loading.

isolated bridges was presented in this work. To this end, a closed form BIRF model is implemented for use in coupled soil-foundation-structure dynamic SSI analysis. Coupled closed form BIRF–FEM numerical models of a physical problem were subjected to an equivalent force time history from the acceleration record of El Centro 1940. Rigid base analysis was used as a baseline to illustrate the effects of considering SSI. The numerical model was also analyzed in an isolated and un-isolated form in order to investigate the effects of SSI on the effectiveness of seismic isolation retrofit schemes. For the physical model considered, overall SSI effects have been identified as (i) elongation of damped period of vibration, (ii) increased relative pier displacements, (iii) increased composite damping ratio for the unisolated soil-foundation-structure systems, and (iv) de-

creased composite damping ratio for the isolated soilfoundation-structure systems. All of these identified effects are amplified for softer soils such as silty sands, and less pronounced for moderate to very stiff soils. The effects of considering SSI in the analysis of seismically isolated structures are shown to reduce the effectiveness of the isolation system of reducing the demands on the isolated structure for moderate to soft soils. Also the importance of relative rigidity of the complete soil-foundation-structure system is investigated through the use of the natural frequency ratio of the structure to the soil-foundation system. The isolation systems are shown to be effective for relatively low natural frequency ratios (stiffer soils) while ineffective for frequency ratios for moderate to soft soils. A recommendation based on the results of this work was

Please cite this article as: Stehmeyer EH, Rizos DC. Considering dynamic soil structure interaction (SSI) effects on seismic isolation retrofit efficiency and the importance of natural frequency.... Soil Dyn Earthquake Eng (2007), doi:10.1016/j.soildyn.2007.07.008

ARTICLE IN PRESS 12

E.H. Stehmeyer, D.C. Rizos / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]

suggested such that an evaluation of the effects of SSI on seismically isolated structures should be performed and incorporated into the design process for seismic isolation retrofit of structures.

References [1] Imbsen RA. Use of isolation for seismic retrofitting bridges. J Bridge Eng 2001;6(6):425–38. [2] Priestly MJN, Seible F, Calvi GM. Seismic design and retrofit of bridges. New York: Wiley; 1996. [3] Kikuchi M, Aiken ID. An analytical hysteresis model for elastomeric seismic isolation bearings. Earthquake Eng Struct Dyn 1997;26: 215–31. [4] Nagarajaiah S, Ferrell K. Stability of elastomeric seismic isolation bearings. J Struct Eng 1999;125(9):946–54. [5] Zheng J, Takeda T. Effects of soil-structure interaction on seismic response of PC cable-stayed bridge. Soil Dyn Earthquake Eng 1995;14:427–37. [6] Crouse CB, McGuire J. Energy dissipation in soil-structure interaction. Earthquake Spectra 2001;17(2):235–59. [7] Vlassis AG, Spyrakos CC. Seismically isolated bridge piers on shallow soil stratum with soil-structure interaction. Comput Struct 2001;79:2847–61. [8] Tongaonkar NP, Jangid RS. Seismic response of isolated bridges with soil-structure interaction. Soil Dyn Earthquake Eng 2003;23:287–302. [9] Menglin L, Jingning W. Effects of soil-structure interaction on structural vibration control. In: Wolf JP, et al., editors. Dynamic soil structure interaction. Amsterdam: Elsevier; 1998. p. 189–202. [10] Beskos DE. Boundary element methods in dynamic analysis: Part II 1986–1996. ASME Appl Mech Rev 1997;50(3):149–97.

[11] Stehmeyer III EH. Computational simulations of linear soil-foundation-structure systems under dynamic and seismic loading. MS thesis, Department of Civil Engineering, University of South Carolina; 2003. [12] Stehmeyer III EH, Rizos DC. B-spline impulse response functions (BIRF) for transient SSI analysis of rigid foundations. Soil Dyn Earthquake Eng 2006;26:421–34. [13] Wolf JP. Foundation vibration analysis using simple physical models. New Jersey: Prentice-Hall; 1994. [14] Rizos DC, Wang Z. Coupled BEM–FEM solutions for direct time domain soil-structure interaction analysis. Eng Anal Boundary Elem 2002;26:877–88. [15] O’Brien J, Rizos DC. A 3D BEM–FEM methodology for simulation of high speed train induced vibrations. Soil Dyn Earthquake Eng 2005;25:289–301. [16] Bathe KJ. Finite element procedures. New Jersey: Prentice-Hall; 1996. [17] Chopra AK. Dynamics of structures: theory & applications to earthquake engineering. 2nd ed. New Jersey: Prentice-Hall; 2001. [18] Luco JE, Wong HL, Mita A. Active control of the seismic response of structures by combined use of base isolation and absorbing boundaries. Earthquake Eng Struct Dyn 1992;21:525–41. [19] Savange I, Eddy JC, Orsolimi GI. Seismic analysis and base isolation retrofit of a steel truss vertical lift bridge. Comput Struct 1999;72:317–27. [20] Abrahamson E, Mitchell S. Seismic response modification device elements for bridge structures development and verification. Comput Struct 2003;81:463–7. [21] Ali HM, Abdel-Ghaffar AM. Modeling of rubber and lead passivecontrol bearings for seismic analysis. J Struct Eng 1995;121(7): 1134–44. [22] Jangid RS. Seismic response of isolated bridges. J Bridge Eng 2004;9(2):156–66. [23] Kunde MC, Jangid RS. Effects of pier and deck flexibility on the seismic response of isolated bridges. J Bridge Eng 2006;11(1):109–21.

Please cite this article as: Stehmeyer EH, Rizos DC. Considering dynamic soil structure interaction (SSI) effects on seismic isolation retrofit efficiency and the importance of natural frequency.... Soil Dyn Earthquake Eng (2007), doi:10.1016/j.soildyn.2007.07.008