- Email: [email protected]
Boundary reaction method for nonlinear analysis of soil–structure interaction under earthquake loads
Soil Dynamics and Earthquake Engineering 89 (2016) 85–90
Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
Boundary reaction method for nonlinear analysis of soil–structure interaction under earthquake loads Jae-min Kim a,n, Eun-haeng Lee b, Sang-hoon Lee c a
Department of Marine and Civil Engineering, Chonnam National University, Yeosu, Jeollanam-do 550-749, Republic of Korea Department of Civil and Environmental Engineering, Graduate School, Chonnam National University, Yeosu, Jeollanam-do 550-749, Republic of Korea c KEPCO E&C, Gimcheon, Gyeongsangbuk-do, Republic of Korea b
art ic l e i nf o
a b s t r a c t
Article history: Received 30 May 2016 Received in revised form 26 July 2016 Accepted 27 July 2016
This paper presents a boundary reaction method (BRM) for nonlinear time domain analysis of soil– structure interaction (SSI) under incident seismic waves. The BRM is a hybrid frequency–time domain method, but it removes global iterations between frequency and time domain analyses commonly required in the hybrid approach, so that it operates as a two-step uncoupled method. Specifically, the nonlinear SSI system is represented as a simple summation of two substructures as follows: (I) wave scattering substructure subjected to incident seismic waves to calculate boundary reaction forces on the fixed interface boundary between a finite nonlinear structure-soil body and an unbounded linear domain; and (II) wave radiation substructure subjected to the boundary reaction forces in which the nonlinearities can be considered. The nonlinear responses in the structure–soil body can be obtained by solving the radiation problem in the time domain using a general-purpose nonlinear finite element code that can simulate absorbing boundary conditions, while the boundary reaction forces can be easily calculated by solving the linear scattering problem by means of a standard frequency domain SSI code. The BRM is verified by comparing the numerical results obtained by the proposed BRM and the conventional frequency-domain SSI analysis for an equivalent linear SSI system. Finally, the BRM is applied to the nonlinear time-domain seismic analysis of a base-isolated nuclear power plant structure supported by a layered soil medium. The numerical results showed that the proposed method is very effective for nonlinear time-domain SSI analyses of nonlinear structure-soil system subjected to earthquake loadings. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear soil–structure interaction BRM (boundary reaction method) Base-isolated nuclear power plant Hybrid frequency–time domain method Nonlinear soil
1. Introduction Generally, a soil–structure interaction (SSI) analysis finds a solution in the frequency domain to consider the radiation of elastic waves transmitted to infinite region [1]. Because the analysis methods in the frequency domain commonly use the superposition principle, linear elasticity is assumed for the structure and the soil. Accordingly, a typical SSI analysis program used to analyze the viscoelastic behavior seems to be inadequate for direct application in nonlinear SSI analyses such as seismic analysis of baseisolated nuclear power plant structures [2]. To consider nonlinear behavior in an SSI analysis, the equivalent linear frequency-domain analysis [3–5], hybrid frequency– time domain (HFTD) analysis [6], and hybrid time–frequency domain (HTFD) analysis [7] have widely been employed as well as time domain techniques such as direct approaches [8] and n
Corresponding author. E-mail address: [email protected] (J.-m. Kim).
http://dx.doi.org/10.1016/j.soildyn.2016.07.020 0267-7261/& 2016 Elsevier Ltd. All rights reserved.
substructure methods [9–11]. However, both hybrid approaches cannot derive the numerical results at once, the HFTD method repeats the analysis until the error in the nonlinear restoring force is reduced and the HTFD method reiterates the analysis until the error in the interacting force becomes small. Likewise, direct procedure has a lot of degrees of freedom in the soil region resulting in significant running time. Recently, a method concurrently applying the domain reduction method (DRM) [10] and the perfectly matched layer (PML) [11,12] has been developed. However, there are still very few general-purpose finite element analysis programs providing such nonlinear SSI analysis function. This paper proposes a boundary reaction method (BRM) based on the fixed boundary wave input method [4] as an uncoupled HTFD method, which operates in two steps without any need for iterative process and enables simultaneous consideration of the nonlinearity and the SSI. The BRM combines a linear SSI analysis in the frequency domain with a time domain finite element analysis program capable of executing the nonlinear analysis so as to conduct the nonlinear SSI analysis. The BRM is verified by comparing the numerical results obtained by the proposed BRM and the
86
J.-m. Kim et al. / Soil Dynamics and Earthquake Engineering 89 (2016) 85–90
conventional frequency-domain SSI analysis for an equivalent linear SSI system. Finally, the proposed BRM is applied for the nonlinear SSI analysis of a base-isolated nuclear power plant (NPP) structure to demonstrate the accuracy and effectiveness of the method.
2. Boundary reaction method for nonlinear SSI analysis Let us consider a structure–soil body that behaves nonlinearly and is supported by a linear elastic near- and far-field bodies subjected to incident seismic waves as shown in Fig. 1(a). If the nonlinear structure–soil body is separated from the linear soil parts, the corresponding equation of motion can be expressed as
⎧ u¨ t t ⎫ ⎧ f NL ( t )⎫ ⎡ 0 0 ⎤⎪ s ( )⎪ ⎪ ⎢0 0 ⎪ s ⎥⎪ ¨ t ⎪ ⎪ f NL t ⎪ ⎢ 0 Mbb Mbn 0 ⎥ ⎪ ub ( t ) ⎪ ⎨ ⎬ ⎨ b ( )⎬ + ⎪ 0 ⎪ ⎢ 0 Mnb Mnn Mne ⎥ ⎪ u¨ tn ( t )⎪ ⎥ ⎢ ⎪ t ⎪ ⎪ ⎪ ⎩ 0 ⎭ ⎣ 0 0 M en M ee ⎦ ⎪ ⎩ u¨ e ( t ) ⎪ ⎭ ⎧ ut t ⎫ ⎡0 0 0 0 ⎤⎪ s ( )⎪ ⎧ 0 ⎫ ⎪ ⎥⎪ t ⎪ ⎪ ⎢ 0 K bb K bn 0 ⎥ ⎪ ub ( t ) ⎪ ⎪ 0 ⎪ ⎬=⎨ ⎨ ⎬ +⎢ ⎢ 0 K nb K nn K ne ⎥ ⎪ ut ( t )⎪ ⎪ 0 ⎪ ⎥⎪ n ⎪ ⎪ t ⎪ ⎢ ⎣ 0 0 K en K ee ⎦ ⎪ ut t ⎪ ⎩ r e ( t )⎭ ⎩ e ( )⎭
and K are respectively mass and stiffness matrices in the linear body; the superscript NL stands for the contribution of the nonlinear structure–soil body; and the subscripts s , b , n, and e represent the degrees of freedom at nodes in bodies or on interface boundaries as depicted in Fig. 1(a). Note that the damping force in the linear region is neglected in Eq. (1) for simplicity. If the nonlinear force vector f NL ( t ) in Eq. (1) is approximated as a linear combination of the total response vectors ( ut ( t ), u̇ t ( t ), and u¨ t ( t )), the solution of Eq. (1) can be obtained by standard SSI techniques, such as boundary element method [13], and finite element method coupled with transmitting boundary element [3] or with dynamic infinite element [4,5], either in the frequency domain or in the time domain. The nonlinear SSI problem in Fig. 1(a) can be described by the superposition of substructure (I), whose nodes on the nonlinear– linear interface are fixed as shown in Fig. 1(b), and substructure (II) of Fig. 1(c), which is defined as a subtraction of the substructure (I) from the nonlinear SSI problem in Fig. 1(a). Thus, the total solution can be obtained by simple summation as
ut ( t ) = ui ( t ) + uii ( t ) (1)
where f NL ( t ) is a dynamic resultant force vector acting on nodes in the nonlinear structure–soil region including inertia force, damping force, nonlinear restoring force, and so on; ut ( t ), u̇ t ( t ), and u¨ t ( t ) are respectively total displacement, velocity, and acceleration vectors as measured from a fixed reference; r t ( t ) denotes the interacting force vector at the interface of two adjacent bodies; M
(2)
where the superscripts i and ii stand for the substructure (I) and the substructure (II), respectively, and the motions within the nonlinear body in the substructure (I) are zero, i.e.,
uis ( t ) = 0
and
uib ( t ) = 0
(3)
As a result of the superposition of the two substructures, the equations of motion for the substructures can be expressed as
Fig. 1. Concept of boundary reaction method proposed in this study.
J.-m. Kim et al. / Soil Dynamics and Earthquake Engineering 89 (2016) 85–90
87
Fig. 2. Base-isolated NPP structure example and input motion at ground surface of free-field soil.
⎧ ⎫ ⎡ Mbb Mbn 0 ⎤ ⎪ u¨ ib ( t ) ⎪ ⎡ Kbb Kbn 0 ⎤ ⎧ u ib (t ) ⎫ ⎧ − p L + G ( t )⎫ b ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥⎪ i ⎥⎪ ⎢ ⎬ 0 ⎢ Mnb Mnn Mne ⎥ ⎨ u¨ n ( t )⎬ + ⎢ Knb Knn Kne ⎥ ⎨ u in (t )⎬ = ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎢⎣ 0 M i ⎪ ⎪ i i ⎣⎢ 0 K en K ee ⎦⎥ ⎪ en M ee ⎦ ⎪ u ⎭ ⎩ re( t ) ⎭ ⎩ u e (t ) ⎪ ⎩ ¨ e( t)⎪ ⎭
also valid for the interaction forces on the outer boundary of the finite part of the free-field soil, i.e.,
(4)
r te ( t ) = r ie ( t ) + r iie ( t )
⎧ uii t ⎫ ⎧ u¨ ii t ⎫ () ⎧ 0 ⎫ () ⎧ f NL ( t )⎫ ⎡ 0 0 ⎤⎪ s ⎪ ⎡ 0 0 0 0 ⎤⎪ s ⎪ ⎪ ⎪ ⎪ ⎢0 0 ⎪ s ⎥ ⎪ ii ⎪ ⎥ ⎪ ¨ ii ⎪ ⎢ t u u ⎪ f NL t ⎪ ⎢ 0 Mbb Mbn 0 ⎥ ⎪ b ( ) ⎪ ⎢ 0 K bb K bn 0 ⎥ ⎪ b ( t ) ⎪ ⎪ p bL+ G ( t )⎪ ⎨ ⎬=⎨ ⎬ ⎨ ⎬+ ⎨ b ( )⎬ + 0 ⎪ ⎪ 0 ⎪ ⎢ 0 Mnb Mnn Mne ⎥ ⎪ u¨ iin ( t ) ⎪ ⎢ 0 K nb K nn K ne ⎥ ⎪ uiin ( t ) ⎪ ⎪ ⎥ ⎢ ⎥ ⎢ ⎪ ⎪ ⎪ ii ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 ⎭ ⎣ 0 0 M en M ee ⎦ ⎪ u¨ ii ( t )⎪ ⎣ 0 0 K en K ee ⎦ ⎪ uii ( t )⎪ ⎩ r e ( t ) ⎭ ⎩ e ⎭ ⎩ e ⎭ where
pbL+ G ( t ) is the reaction force vector at nodes on the
boundary b due to actions in the linear body ( L ), which includes irregular near-field soil, finite part of free-field soil, and unbounded part of the free-field soil ( G) as shown in Fig. 1(b). It is important to note that the substructure (I) constitutes a linear scattering problem where the effective seismic load at the fixed interface can be calculated readily in the frequency domain, and the substructure (II) presents a wave radiation problem for which a nonlinear forced vibration analysis can be performed effectively in the time domain. In addition, the principle of superposition is
(6)
(5)
The equivalence of the solution obtained by simple summation of the two substructures to that in the total nonlinear problem can be easily proven by showing that the substitution of (Eqs. (2) and 3) into Eq. (5) and application of relations in (Eqs. (4) and 6) lead to the equation of motion for the total system given in Eq. (1). As can be seen from (Eqs. (2) and 3), the solution within the nonlinear structure–soil body in the substructure (II) is identical to ubii ( t ) = utb ( t ). that in the total responses, i.e., uiis ( t ) = uts ( t ) and Thus, the nonlinear resultant force vector f NL ( t ) risen in the total system is the same as that in the substructure (II).
88
J.-m. Kim et al. / Soil Dynamics and Earthquake Engineering 89 (2016) 85–90
Fig. 3. Numerical model for nonlinear SSI analysis by the proposed BRM.
J.-m. Kim et al. / Soil Dynamics and Earthquake Engineering 89 (2016) 85–90
In summary, the proposed BRM carries out the nonlinear SSI analysis in two steps and, unlike the conventional HTFD method [7], presents the advantage of not needing to repeat time-domain analysis and frequency-domain analysis. At this juncture, it is notable that effect of the nonlinear body in the BRM, as can be seen from Eq. (5), are completely decomposed from that of the linear parts by extending the rigid boundary substructure concept, while they are partially coupled in the flexible boundary substructure technique [9]. To apply the BRM technique, a program capable of conducting linear SSI analysis is needed for the calculation of reaction forces at the fixed interface boundary in the BRM. Moreover, the generalpurpose finite element program used in the BRM provides a nonlinear analysis function and a capability of efficiently simulating the wave radiation conditions at the outer boundary. The absorbing boundary conditions (ABCs) supported by the generalpurpose finite element programs are the viscous damper element, the viscous damper and spring elements [8,14], and the PML [11].
89
Numerical model for nonlinear SSI analysis by the proposed BRM is shown in Fig. 3. The nonlinear behavior of soil shall be considered in dynamic analyses of NPP structures and may be approximated by equivalent linear material properties [15]. In this study, the primary nonlinearity in the free-field soil is considered in an equivalent linear manner according to ASCE 4-98 [15], while the secondary nonlinearity is considered by means of the DruckerPrager plastic model [12,16]. The soil region subjected to nonlinear behavior is depicted in Fig. 3(a). The seismic input is assumed as elastic waves excited by vertically incident SV waves. The equivalent linear properties of the soil layers that are determined by iterative free-field analysis utilizing the SHAKE program are plotted in Fig. 3(a). The structure and near-field soil are modeled for the BRM analysis of the wave radiation problem using ANSYS [16]. Spring elements and viscous damping elements are disposed at the outer boundary of the truncated finite element region [8,14]. The spring and the damping coefficients per unit area of boundary 1 are calculated as: kBN = 2r ρVP2, cBN = ρVP (normal direction); 1
3. Numerical examples The base-isolated NPP model shown in Fig. 2 is taken as the numerical example. This model is based on the NPP model included in the SASSI manual [3]. The design response spectrum of the input motion is based on the RG 1.60 design response spectrum but augmented in high-frequency ranges. The input displacement–time history is generated to match the design response spectrum and input as horizontal motion at the ground surface of the free-field soil.
3.1. Equivalent linear SSI problem
1.5
Spectral Acceleration (g)
kBT = 2r ρVS2, cBT = ρVS (tangential directions), where r ¼radial distance from the center of the foundation to the finite element boundary; ρ ¼ mass density of soil; VP ¼P-wave velocity of soil; and VS ¼S-wave velocity of soil. The reaction forces at the boundary for the BRM analysis in the linear SSI problem are calculated by conventional SSI analysis in which the nodes on the boundary b are fixed. The seismic waves in the free-field are assumed as vertically incident SV waves, and the KIESSI-3D model shown in Fig. 3 (b) is used to calculate the reaction forces. Typical time histories of the reaction forces at nodes R1, R2, and R3 are plotted.
Equivalent Linear Frequency Domain SSI (KIESSI-3D) Equvalent Linear BRM (KIESSI-3D & ANSYS)
5% Damping Ratio 1
0.5
0 0.1
1
Frequency (Hz)
10
100
Fig. 4. Response spectra for horizontal response at the top of containment of the base-isolated NPP example by equivalent linear SSI analysis.
Any numerical procedure for the nonlinear SSI analysis is required to verify by comparing responses of a linearized model for the nonlinear SSI problem with those by conventional frequencydomain SSI analysis. In this study, a linear time-domain analysis for an equivalent linear model is conducted to verify the BRM numerical model. The verification is done using through a comparison with the structural responses obtained by means of conventional SSI seismic analysis using the KIESSI-3D program [4,5] and finite element mesh in Fig. 3(a). Fig. 4 compares the horizontal acceleration response spectra at the top of the containment building. The accuracy of the BRM numerical model for the nonlinear analysis is verified by the good agreement between the linear analysis results by BRM and the linear SSI analysis results in the frequency domain for each of the considered soil profiles.
Fig. 5. The plastic strain of the near field soil inside the boundary b obtained through the nonlinear SSI analysis by BRM.
90
J.-m. Kim et al. / Soil Dynamics and Earthquake Engineering 89 (2016) 85–90
3.2. Nonlinear SSI problem
Acknowledgment
Nonlinear SSI analysis is performed for the base-isolated structure and the near field soil inside the boundary bby using the analysis model and the calibrated boundary reaction forces as shown in Fig. 3(b). The BRM analysis model for the nonlinear SSI analysis is identical to the equivalent linear analysis model with the exception of the nonlinear elements of the base isolators and the near field soil inside the boundary b. The nonlinear properties of the base isolators are defined in Fig. 3(a). Besides, Extended Drucker-Prager soil model [16] is used to consider the nonlinear behavior of the near field soil inside the boundary b . The material properties for Extended Drucker-Prager soil model are as follows: Cohesion value c ¼1.321 MPa, friction angle θ = 30.0° , dilation angle θf = 25.0° for soil layer from ground level (GL) to GL-6.0 m, and c ¼1.382 MPa, θ = 30.0° , θf = 25.0° for soil layer between GL6.0 m and GL-12.0 m. In this study, the nonlinear hysteresis of the base isolator and the stress-strain curves of the near field soil inside the boundary b obtained by the nonlinear SSI analysis using BRM are observed. The displacement–force hysteresis curve of the base isolator is shown in Fig. 3(a). Finally, the plastic strain of the near field soil inside the boundary b obtained by the nonlinear SSI analysis of the base-isolated system is shown in Fig. 5(a) in comparison with that of non-isolated system as depicted in Fig. 5(b). This comparison indicates that the base-isolated system can suppress nonlinear behavior of the soil in the vicinity of basemat as well as reduce acceleration of the superstructure.
This study was supported by the Nuclear Research & Development project of the KETEP through a grant (2014151010170A) funded by the Ministry of Knowledge Economy, Republic of Korea, and also supported by a grant (14CTAP-C077514-01) from Technology Advancement Research Program funded by Ministry of Land, Infrastructure and Transport of Korean government. They are gratefully acknowledged.
4. Conclusions This paper proposed the BRM (boundary reaction method) as a two-step uncoupled hybrid time–frequency domain method for nonlinear SSI analyses. The BRM was verified for an SSI problem by comparing the numerical result obtained by the proposed BRM and the conventional frequency domain SSI analysis for an equivalent linear SSI system. Finally, the proposed BRM was applied for the nonlinear SSI analysis of a base-isolated NPP structure to demonstrate the effectiveness of the method. The results showed that the proposed BRM can be effectively applied for nonlinear SSI analyses.
References [1] Wolf JP. Dynamic soil-structure interaction analysis. Prentice-Hall; 1985. [2] Coleman J, Jeremic B, Whittaker A. Nonlinear time domain seismic soil structure interaction (SSI) Analysis for Nuclear facilities and draft appendix b of Asce 4. SMiRT-22, August 18–23, San Francisco, CA; 2013. [3] Lysmer J, Tabatabaie-Raissi M, Tajirian F, Vahdani S, Ostadan F. SASSI: a system for analysis of soil-structure interaction – user's manual.Berkeley, CA: University of California; 1988. [4] Ryu JS, Seo CG, Kim JM, Yun CB. Seismic response analysis of soil–structure interactive system using a coupled three-dimensional FE–IE method. Nucl Eng Des 2010:1949–66. [5] Lee EH, Kim JM, Seo CG. Large-scale 3D SSI analysis using KIESSI-3D program. COSEIK J Comput Struct Eng 2013;26:439–45. [6] Kawamoto JD. Solution of nonlinear dynamic structural system based on a hybrid frequency-time-domain approach. USA: MIT; 1983 Research report R83-5. [7] Bernal D, Youssef A. A hybrid time frequency domain formulation for nonlinear soil-structure interaction. Earthq Eng Struct Dyn 1998;27:673–85. [8] Jingbo L, Yandong L. A direct method for analysis of dynamic soil-structure interaction based on interface idea, developments in geotechnical engineering. In: Zhang C, Wolf JP, editors. Dynamic soil-structure interaction: current research in China and Switzerland. Elsevier; 1998. p. 261–76. [9] Bayo E, Wilson EL. Solution of the three dimensional soil-structure interaction problem in the time domain.. San Francisco, CA: 8th WCEE; 1984. p. 961–8. [10] Bielak J, Loukakis K, Hisada Y, Yoshimura C. Domain reduction method for three–dimensional earthquake modeling in localized regions. part I: theory. Bull Seismol Soc Am 2003;93(2):817–24. [11] Basu U. Explicit finite element perfectly matched layer for transient threedimensional elastic waves. Int J Numer Methods Eng 2009;77:151–76. [12] Lee JH, Kim JH, Kim JK. Perfectly matched discrete layers for three-dimensional nonlinear soil-structure interaction analysis. Comput Struct 2016;165:34–47. [13] Karabalis DL, Beskos DE. Dynamic response of 3-D rigid surface foundations by time-domain boundary element method. Earthq Eng Struct Dyn 1984;12:73– 93. [14] Kellezi L. Local transmitting boundaries for transient elastic analysis. Soil Dyn Earthq Eng 2000;19:533–47. [15] ASCE 4-98. Seismic analysis of safety-related nuclear structures and commentary. ASCE; 1999. [16] ANSYS Inc. ANSYS mechanical user's guide, Release 15.0; 2013.
Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
Boundary reaction method for nonlinear analysis of soil–structure interaction under earthquake loads Jae-min Kim a,n, Eun-haeng Lee b, Sang-hoon Lee c a
Department of Marine and Civil Engineering, Chonnam National University, Yeosu, Jeollanam-do 550-749, Republic of Korea Department of Civil and Environmental Engineering, Graduate School, Chonnam National University, Yeosu, Jeollanam-do 550-749, Republic of Korea c KEPCO E&C, Gimcheon, Gyeongsangbuk-do, Republic of Korea b
art ic l e i nf o
a b s t r a c t
Article history: Received 30 May 2016 Received in revised form 26 July 2016 Accepted 27 July 2016
This paper presents a boundary reaction method (BRM) for nonlinear time domain analysis of soil– structure interaction (SSI) under incident seismic waves. The BRM is a hybrid frequency–time domain method, but it removes global iterations between frequency and time domain analyses commonly required in the hybrid approach, so that it operates as a two-step uncoupled method. Specifically, the nonlinear SSI system is represented as a simple summation of two substructures as follows: (I) wave scattering substructure subjected to incident seismic waves to calculate boundary reaction forces on the fixed interface boundary between a finite nonlinear structure-soil body and an unbounded linear domain; and (II) wave radiation substructure subjected to the boundary reaction forces in which the nonlinearities can be considered. The nonlinear responses in the structure–soil body can be obtained by solving the radiation problem in the time domain using a general-purpose nonlinear finite element code that can simulate absorbing boundary conditions, while the boundary reaction forces can be easily calculated by solving the linear scattering problem by means of a standard frequency domain SSI code. The BRM is verified by comparing the numerical results obtained by the proposed BRM and the conventional frequency-domain SSI analysis for an equivalent linear SSI system. Finally, the BRM is applied to the nonlinear time-domain seismic analysis of a base-isolated nuclear power plant structure supported by a layered soil medium. The numerical results showed that the proposed method is very effective for nonlinear time-domain SSI analyses of nonlinear structure-soil system subjected to earthquake loadings. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear soil–structure interaction BRM (boundary reaction method) Base-isolated nuclear power plant Hybrid frequency–time domain method Nonlinear soil
1. Introduction Generally, a soil–structure interaction (SSI) analysis finds a solution in the frequency domain to consider the radiation of elastic waves transmitted to infinite region [1]. Because the analysis methods in the frequency domain commonly use the superposition principle, linear elasticity is assumed for the structure and the soil. Accordingly, a typical SSI analysis program used to analyze the viscoelastic behavior seems to be inadequate for direct application in nonlinear SSI analyses such as seismic analysis of baseisolated nuclear power plant structures [2]. To consider nonlinear behavior in an SSI analysis, the equivalent linear frequency-domain analysis [3–5], hybrid frequency– time domain (HFTD) analysis [6], and hybrid time–frequency domain (HTFD) analysis [7] have widely been employed as well as time domain techniques such as direct approaches [8] and n
Corresponding author. E-mail address: [email protected] (J.-m. Kim).
http://dx.doi.org/10.1016/j.soildyn.2016.07.020 0267-7261/& 2016 Elsevier Ltd. All rights reserved.
substructure methods [9–11]. However, both hybrid approaches cannot derive the numerical results at once, the HFTD method repeats the analysis until the error in the nonlinear restoring force is reduced and the HTFD method reiterates the analysis until the error in the interacting force becomes small. Likewise, direct procedure has a lot of degrees of freedom in the soil region resulting in significant running time. Recently, a method concurrently applying the domain reduction method (DRM) [10] and the perfectly matched layer (PML) [11,12] has been developed. However, there are still very few general-purpose finite element analysis programs providing such nonlinear SSI analysis function. This paper proposes a boundary reaction method (BRM) based on the fixed boundary wave input method [4] as an uncoupled HTFD method, which operates in two steps without any need for iterative process and enables simultaneous consideration of the nonlinearity and the SSI. The BRM combines a linear SSI analysis in the frequency domain with a time domain finite element analysis program capable of executing the nonlinear analysis so as to conduct the nonlinear SSI analysis. The BRM is verified by comparing the numerical results obtained by the proposed BRM and the
86
J.-m. Kim et al. / Soil Dynamics and Earthquake Engineering 89 (2016) 85–90
conventional frequency-domain SSI analysis for an equivalent linear SSI system. Finally, the proposed BRM is applied for the nonlinear SSI analysis of a base-isolated nuclear power plant (NPP) structure to demonstrate the accuracy and effectiveness of the method.
2. Boundary reaction method for nonlinear SSI analysis Let us consider a structure–soil body that behaves nonlinearly and is supported by a linear elastic near- and far-field bodies subjected to incident seismic waves as shown in Fig. 1(a). If the nonlinear structure–soil body is separated from the linear soil parts, the corresponding equation of motion can be expressed as
⎧ u¨ t t ⎫ ⎧ f NL ( t )⎫ ⎡ 0 0 ⎤⎪ s ( )⎪ ⎪ ⎢0 0 ⎪ s ⎥⎪ ¨ t ⎪ ⎪ f NL t ⎪ ⎢ 0 Mbb Mbn 0 ⎥ ⎪ ub ( t ) ⎪ ⎨ ⎬ ⎨ b ( )⎬ + ⎪ 0 ⎪ ⎢ 0 Mnb Mnn Mne ⎥ ⎪ u¨ tn ( t )⎪ ⎥ ⎢ ⎪ t ⎪ ⎪ ⎪ ⎩ 0 ⎭ ⎣ 0 0 M en M ee ⎦ ⎪ ⎩ u¨ e ( t ) ⎪ ⎭ ⎧ ut t ⎫ ⎡0 0 0 0 ⎤⎪ s ( )⎪ ⎧ 0 ⎫ ⎪ ⎥⎪ t ⎪ ⎪ ⎢ 0 K bb K bn 0 ⎥ ⎪ ub ( t ) ⎪ ⎪ 0 ⎪ ⎬=⎨ ⎨ ⎬ +⎢ ⎢ 0 K nb K nn K ne ⎥ ⎪ ut ( t )⎪ ⎪ 0 ⎪ ⎥⎪ n ⎪ ⎪ t ⎪ ⎢ ⎣ 0 0 K en K ee ⎦ ⎪ ut t ⎪ ⎩ r e ( t )⎭ ⎩ e ( )⎭
and K are respectively mass and stiffness matrices in the linear body; the superscript NL stands for the contribution of the nonlinear structure–soil body; and the subscripts s , b , n, and e represent the degrees of freedom at nodes in bodies or on interface boundaries as depicted in Fig. 1(a). Note that the damping force in the linear region is neglected in Eq. (1) for simplicity. If the nonlinear force vector f NL ( t ) in Eq. (1) is approximated as a linear combination of the total response vectors ( ut ( t ), u̇ t ( t ), and u¨ t ( t )), the solution of Eq. (1) can be obtained by standard SSI techniques, such as boundary element method [13], and finite element method coupled with transmitting boundary element [3] or with dynamic infinite element [4,5], either in the frequency domain or in the time domain. The nonlinear SSI problem in Fig. 1(a) can be described by the superposition of substructure (I), whose nodes on the nonlinear– linear interface are fixed as shown in Fig. 1(b), and substructure (II) of Fig. 1(c), which is defined as a subtraction of the substructure (I) from the nonlinear SSI problem in Fig. 1(a). Thus, the total solution can be obtained by simple summation as
ut ( t ) = ui ( t ) + uii ( t ) (1)
where f NL ( t ) is a dynamic resultant force vector acting on nodes in the nonlinear structure–soil region including inertia force, damping force, nonlinear restoring force, and so on; ut ( t ), u̇ t ( t ), and u¨ t ( t ) are respectively total displacement, velocity, and acceleration vectors as measured from a fixed reference; r t ( t ) denotes the interacting force vector at the interface of two adjacent bodies; M
(2)
where the superscripts i and ii stand for the substructure (I) and the substructure (II), respectively, and the motions within the nonlinear body in the substructure (I) are zero, i.e.,
uis ( t ) = 0
and
uib ( t ) = 0
(3)
As a result of the superposition of the two substructures, the equations of motion for the substructures can be expressed as
Fig. 1. Concept of boundary reaction method proposed in this study.
J.-m. Kim et al. / Soil Dynamics and Earthquake Engineering 89 (2016) 85–90
87
Fig. 2. Base-isolated NPP structure example and input motion at ground surface of free-field soil.
⎧ ⎫ ⎡ Mbb Mbn 0 ⎤ ⎪ u¨ ib ( t ) ⎪ ⎡ Kbb Kbn 0 ⎤ ⎧ u ib (t ) ⎫ ⎧ − p L + G ( t )⎫ b ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥⎪ i ⎥⎪ ⎢ ⎬ 0 ⎢ Mnb Mnn Mne ⎥ ⎨ u¨ n ( t )⎬ + ⎢ Knb Knn Kne ⎥ ⎨ u in (t )⎬ = ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎢⎣ 0 M i ⎪ ⎪ i i ⎣⎢ 0 K en K ee ⎦⎥ ⎪ en M ee ⎦ ⎪ u ⎭ ⎩ re( t ) ⎭ ⎩ u e (t ) ⎪ ⎩ ¨ e( t)⎪ ⎭
also valid for the interaction forces on the outer boundary of the finite part of the free-field soil, i.e.,
(4)
r te ( t ) = r ie ( t ) + r iie ( t )
⎧ uii t ⎫ ⎧ u¨ ii t ⎫ () ⎧ 0 ⎫ () ⎧ f NL ( t )⎫ ⎡ 0 0 ⎤⎪ s ⎪ ⎡ 0 0 0 0 ⎤⎪ s ⎪ ⎪ ⎪ ⎪ ⎢0 0 ⎪ s ⎥ ⎪ ii ⎪ ⎥ ⎪ ¨ ii ⎪ ⎢ t u u ⎪ f NL t ⎪ ⎢ 0 Mbb Mbn 0 ⎥ ⎪ b ( ) ⎪ ⎢ 0 K bb K bn 0 ⎥ ⎪ b ( t ) ⎪ ⎪ p bL+ G ( t )⎪ ⎨ ⎬=⎨ ⎬ ⎨ ⎬+ ⎨ b ( )⎬ + 0 ⎪ ⎪ 0 ⎪ ⎢ 0 Mnb Mnn Mne ⎥ ⎪ u¨ iin ( t ) ⎪ ⎢ 0 K nb K nn K ne ⎥ ⎪ uiin ( t ) ⎪ ⎪ ⎥ ⎢ ⎥ ⎢ ⎪ ⎪ ⎪ ii ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 ⎭ ⎣ 0 0 M en M ee ⎦ ⎪ u¨ ii ( t )⎪ ⎣ 0 0 K en K ee ⎦ ⎪ uii ( t )⎪ ⎩ r e ( t ) ⎭ ⎩ e ⎭ ⎩ e ⎭ where
pbL+ G ( t ) is the reaction force vector at nodes on the
boundary b due to actions in the linear body ( L ), which includes irregular near-field soil, finite part of free-field soil, and unbounded part of the free-field soil ( G) as shown in Fig. 1(b). It is important to note that the substructure (I) constitutes a linear scattering problem where the effective seismic load at the fixed interface can be calculated readily in the frequency domain, and the substructure (II) presents a wave radiation problem for which a nonlinear forced vibration analysis can be performed effectively in the time domain. In addition, the principle of superposition is
(6)
(5)
The equivalence of the solution obtained by simple summation of the two substructures to that in the total nonlinear problem can be easily proven by showing that the substitution of (Eqs. (2) and 3) into Eq. (5) and application of relations in (Eqs. (4) and 6) lead to the equation of motion for the total system given in Eq. (1). As can be seen from (Eqs. (2) and 3), the solution within the nonlinear structure–soil body in the substructure (II) is identical to ubii ( t ) = utb ( t ). that in the total responses, i.e., uiis ( t ) = uts ( t ) and Thus, the nonlinear resultant force vector f NL ( t ) risen in the total system is the same as that in the substructure (II).
88
J.-m. Kim et al. / Soil Dynamics and Earthquake Engineering 89 (2016) 85–90
Fig. 3. Numerical model for nonlinear SSI analysis by the proposed BRM.
J.-m. Kim et al. / Soil Dynamics and Earthquake Engineering 89 (2016) 85–90
In summary, the proposed BRM carries out the nonlinear SSI analysis in two steps and, unlike the conventional HTFD method [7], presents the advantage of not needing to repeat time-domain analysis and frequency-domain analysis. At this juncture, it is notable that effect of the nonlinear body in the BRM, as can be seen from Eq. (5), are completely decomposed from that of the linear parts by extending the rigid boundary substructure concept, while they are partially coupled in the flexible boundary substructure technique [9]. To apply the BRM technique, a program capable of conducting linear SSI analysis is needed for the calculation of reaction forces at the fixed interface boundary in the BRM. Moreover, the generalpurpose finite element program used in the BRM provides a nonlinear analysis function and a capability of efficiently simulating the wave radiation conditions at the outer boundary. The absorbing boundary conditions (ABCs) supported by the generalpurpose finite element programs are the viscous damper element, the viscous damper and spring elements [8,14], and the PML [11].
89
Numerical model for nonlinear SSI analysis by the proposed BRM is shown in Fig. 3. The nonlinear behavior of soil shall be considered in dynamic analyses of NPP structures and may be approximated by equivalent linear material properties [15]. In this study, the primary nonlinearity in the free-field soil is considered in an equivalent linear manner according to ASCE 4-98 [15], while the secondary nonlinearity is considered by means of the DruckerPrager plastic model [12,16]. The soil region subjected to nonlinear behavior is depicted in Fig. 3(a). The seismic input is assumed as elastic waves excited by vertically incident SV waves. The equivalent linear properties of the soil layers that are determined by iterative free-field analysis utilizing the SHAKE program are plotted in Fig. 3(a). The structure and near-field soil are modeled for the BRM analysis of the wave radiation problem using ANSYS [16]. Spring elements and viscous damping elements are disposed at the outer boundary of the truncated finite element region [8,14]. The spring and the damping coefficients per unit area of boundary 1 are calculated as: kBN = 2r ρVP2, cBN = ρVP (normal direction); 1
3. Numerical examples The base-isolated NPP model shown in Fig. 2 is taken as the numerical example. This model is based on the NPP model included in the SASSI manual [3]. The design response spectrum of the input motion is based on the RG 1.60 design response spectrum but augmented in high-frequency ranges. The input displacement–time history is generated to match the design response spectrum and input as horizontal motion at the ground surface of the free-field soil.
3.1. Equivalent linear SSI problem
1.5
Spectral Acceleration (g)
kBT = 2r ρVS2, cBT = ρVS (tangential directions), where r ¼radial distance from the center of the foundation to the finite element boundary; ρ ¼ mass density of soil; VP ¼P-wave velocity of soil; and VS ¼S-wave velocity of soil. The reaction forces at the boundary for the BRM analysis in the linear SSI problem are calculated by conventional SSI analysis in which the nodes on the boundary b are fixed. The seismic waves in the free-field are assumed as vertically incident SV waves, and the KIESSI-3D model shown in Fig. 3 (b) is used to calculate the reaction forces. Typical time histories of the reaction forces at nodes R1, R2, and R3 are plotted.
Equivalent Linear Frequency Domain SSI (KIESSI-3D) Equvalent Linear BRM (KIESSI-3D & ANSYS)
5% Damping Ratio 1
0.5
0 0.1
1
Frequency (Hz)
10
100
Fig. 4. Response spectra for horizontal response at the top of containment of the base-isolated NPP example by equivalent linear SSI analysis.
Any numerical procedure for the nonlinear SSI analysis is required to verify by comparing responses of a linearized model for the nonlinear SSI problem with those by conventional frequencydomain SSI analysis. In this study, a linear time-domain analysis for an equivalent linear model is conducted to verify the BRM numerical model. The verification is done using through a comparison with the structural responses obtained by means of conventional SSI seismic analysis using the KIESSI-3D program [4,5] and finite element mesh in Fig. 3(a). Fig. 4 compares the horizontal acceleration response spectra at the top of the containment building. The accuracy of the BRM numerical model for the nonlinear analysis is verified by the good agreement between the linear analysis results by BRM and the linear SSI analysis results in the frequency domain for each of the considered soil profiles.
Fig. 5. The plastic strain of the near field soil inside the boundary b obtained through the nonlinear SSI analysis by BRM.
90
J.-m. Kim et al. / Soil Dynamics and Earthquake Engineering 89 (2016) 85–90
3.2. Nonlinear SSI problem
Acknowledgment
Nonlinear SSI analysis is performed for the base-isolated structure and the near field soil inside the boundary bby using the analysis model and the calibrated boundary reaction forces as shown in Fig. 3(b). The BRM analysis model for the nonlinear SSI analysis is identical to the equivalent linear analysis model with the exception of the nonlinear elements of the base isolators and the near field soil inside the boundary b. The nonlinear properties of the base isolators are defined in Fig. 3(a). Besides, Extended Drucker-Prager soil model [16] is used to consider the nonlinear behavior of the near field soil inside the boundary b . The material properties for Extended Drucker-Prager soil model are as follows: Cohesion value c ¼1.321 MPa, friction angle θ = 30.0° , dilation angle θf = 25.0° for soil layer from ground level (GL) to GL-6.0 m, and c ¼1.382 MPa, θ = 30.0° , θf = 25.0° for soil layer between GL6.0 m and GL-12.0 m. In this study, the nonlinear hysteresis of the base isolator and the stress-strain curves of the near field soil inside the boundary b obtained by the nonlinear SSI analysis using BRM are observed. The displacement–force hysteresis curve of the base isolator is shown in Fig. 3(a). Finally, the plastic strain of the near field soil inside the boundary b obtained by the nonlinear SSI analysis of the base-isolated system is shown in Fig. 5(a) in comparison with that of non-isolated system as depicted in Fig. 5(b). This comparison indicates that the base-isolated system can suppress nonlinear behavior of the soil in the vicinity of basemat as well as reduce acceleration of the superstructure.
This study was supported by the Nuclear Research & Development project of the KETEP through a grant (2014151010170A) funded by the Ministry of Knowledge Economy, Republic of Korea, and also supported by a grant (14CTAP-C077514-01) from Technology Advancement Research Program funded by Ministry of Land, Infrastructure and Transport of Korean government. They are gratefully acknowledged.
4. Conclusions This paper proposed the BRM (boundary reaction method) as a two-step uncoupled hybrid time–frequency domain method for nonlinear SSI analyses. The BRM was verified for an SSI problem by comparing the numerical result obtained by the proposed BRM and the conventional frequency domain SSI analysis for an equivalent linear SSI system. Finally, the proposed BRM was applied for the nonlinear SSI analysis of a base-isolated NPP structure to demonstrate the effectiveness of the method. The results showed that the proposed BRM can be effectively applied for nonlinear SSI analyses.
References [1] Wolf JP. Dynamic soil-structure interaction analysis. Prentice-Hall; 1985. [2] Coleman J, Jeremic B, Whittaker A. Nonlinear time domain seismic soil structure interaction (SSI) Analysis for Nuclear facilities and draft appendix b of Asce 4. SMiRT-22, August 18–23, San Francisco, CA; 2013. [3] Lysmer J, Tabatabaie-Raissi M, Tajirian F, Vahdani S, Ostadan F. SASSI: a system for analysis of soil-structure interaction – user's manual.Berkeley, CA: University of California; 1988. [4] Ryu JS, Seo CG, Kim JM, Yun CB. Seismic response analysis of soil–structure interactive system using a coupled three-dimensional FE–IE method. Nucl Eng Des 2010:1949–66. [5] Lee EH, Kim JM, Seo CG. Large-scale 3D SSI analysis using KIESSI-3D program. COSEIK J Comput Struct Eng 2013;26:439–45. [6] Kawamoto JD. Solution of nonlinear dynamic structural system based on a hybrid frequency-time-domain approach. USA: MIT; 1983 Research report R83-5. [7] Bernal D, Youssef A. A hybrid time frequency domain formulation for nonlinear soil-structure interaction. Earthq Eng Struct Dyn 1998;27:673–85. [8] Jingbo L, Yandong L. A direct method for analysis of dynamic soil-structure interaction based on interface idea, developments in geotechnical engineering. In: Zhang C, Wolf JP, editors. Dynamic soil-structure interaction: current research in China and Switzerland. Elsevier; 1998. p. 261–76. [9] Bayo E, Wilson EL. Solution of the three dimensional soil-structure interaction problem in the time domain.. San Francisco, CA: 8th WCEE; 1984. p. 961–8. [10] Bielak J, Loukakis K, Hisada Y, Yoshimura C. Domain reduction method for three–dimensional earthquake modeling in localized regions. part I: theory. Bull Seismol Soc Am 2003;93(2):817–24. [11] Basu U. Explicit finite element perfectly matched layer for transient threedimensional elastic waves. Int J Numer Methods Eng 2009;77:151–76. [12] Lee JH, Kim JH, Kim JK. Perfectly matched discrete layers for three-dimensional nonlinear soil-structure interaction analysis. Comput Struct 2016;165:34–47. [13] Karabalis DL, Beskos DE. Dynamic response of 3-D rigid surface foundations by time-domain boundary element method. Earthq Eng Struct Dyn 1984;12:73– 93. [14] Kellezi L. Local transmitting boundaries for transient elastic analysis. Soil Dyn Earthq Eng 2000;19:533–47. [15] ASCE 4-98. Seismic analysis of safety-related nuclear structures and commentary. ASCE; 1999. [16] ANSYS Inc. ANSYS mechanical user's guide, Release 15.0; 2013.