A time-domain substructuring method for dynamic soilstructure-interaction analysis of arbitrarily shaped foundation systems on heterogeneous media Chanseok Jeong1, Elnaz Esmaeilzadeh Seylabi2, and Ertugrul Taciroglu3 Civil & Environmental Engineering Department, University of California, Los Angeles, Los Angeles, CA 90095-1593
ABSTRACT In order to consider soil-structure-interaction (SSI) effects in the seismic design of infrastructure, the dynamic interactions between the far-field soil domain, the—potentially inelastically behaving—near-field soil domain, and the structure should be taken into account. The far-field domain is semi-infinite unless the bedrock, or a rock outcrop is very near, and thus it can be represented with a reduced-order model in the form of impedance functions. The use of impedance functions in SSI analyses allows the computational cost to be reduced by several orders of magnitude, without compromising the solution accuracy. Moreover, it is now possible to obtain time-domain representations of the inherently frequency-dependent impedance functions through, for example, rational approximations. As such, accurate nonlinear time-history analyses of problems that involve SSI effects can be now carried out in a computationally efficient manner. However, the current catalogue of impedance functions is limited to simple foundation shapes and soil profiles (typically homogeneous). In the present study, we provide a systematic approach with which impedance functions for arbitrarily shaped foundations resting on (or embedded in) heterogeneous soil domains can be obtained. In order to obtain the impedance functions, forward wave propagation analyses are carried out on a high-performance computing platform. The finite element method is employed to account for the arbitrary heterogeneity of soil, as well as different foundation types and geometries. The semi-infinite far-field soil is treated with the Perfectly Matched Layers (PML), which, to date, is considered as the best Wave-Absorbing Boundary Condition (WABC) technique. Practical examples are provided that display pronounced variations in impedance functions with respect to frequency, which illustrate and quantify the importance of using frequency-dependent impedance functions in engineering analyses. 1
Post-doctoral Research Associate (
[email protected]) Graduate Student Researcher (
[email protected]) 3 Associate Professor (
[email protected]) 2
1
1. INTRODUCTION Analysis of dynamic soil-structure-interaction (SSI), induced by earthquake motion, is essential to the seismic design and performance assessment of many types of structures. Neglecting the SSI effects, or their poor estimation, could result in over- or under-estimation of the seismic performance of the structure, which in turn, may compromise safety or result in over-designs. Structure Near-field soil
Adjacent soil
Structure Adjacent soil Interface between adjacent soil and near-field soil
Near-field soil (surrounding the adjacent soil) is replaced by an impedance function [Kimp] of the stiffness matrix form.
Far-field soil
Truncated wave absorbing boundary condition
Seismic excitation
The foundation input motion [ufim]: the free field motion of the interface of the excavated soil (the near-field soil except for the structure and its adjacent soil)
Figure 1. (a) The direct method versus (b) the substructure method.
In general, SSI analyses are conducted numerically by means of either the direct method (Fig. 1(a)) or the substructure method (Fig. 1(b)). In the direct method [Bielak (1984) and Wolf (1985)], the near-field soil as well as the structure and its adjacent soil are discretized, and appropriate Wave-Absorbing-Boundary-Conditions (WABCs) are employed at the remote boundaries to truncate the semi-infinite soil domain. Seismic input motions are applied to the system by either applying accurate “jump conditions” along an arbitrary fictitious interface within the near-field soil [Bielak (1984)] or, less accurately, by directly prescribing the separately calculated free-field displacements along truncated boundaries [Wolf (1985)]. The direct method has the advantage of being directly applicable in nonlinear time domain analyses. However, its computational cost is often too high, and as such, it is not commonly attempted in engineering practice. The substructure method [Wolf (1985), Wolf (1989), Mylonakis (1997)] can be used by defining an appropriate set of impedance functions at the interface of the near-field soil and the adjacent soil (or the foundation). Seismically induced input motions are prescribed along the interface of the near-field domain and the adjacent soil, or directly at the foundation. These motions are typically computed using “site response” analyses, which yield free-field motions, and for certain cases, they are modified to account for the so-called “kinematic” effects [see, for example, Gazetes (1984)]—i.e., the effects of the stiffness and geometry of the foundation on the incoming seismic waves. The substructure method is typically carried out via frequency-domain analyses, which have low computational cost, but are nominally
2
restricted to the linear response regime. The method can also be carried out in the time-domain if the frequency-dependent impedance functions can be transformed to the time domain, using techniques such as rational approximations [Safak, 2006]. To date, impedance functions that are available in open literature are typically for simple soil profiles and rigid foundations. The functions are obtained via analytic or semi-analytic methods. In order to utilize the substructure method for a general SSI problem, it is necessary, among other things, to have the tools to obtain impedance functions for arbitrary soil profiles and foundations of different types (rigid, flexible, surface, embedded, void, etc.). In this paper, we present a procedure to numerically computing the impedance functions for a more general setting wherein the foundation is a rigid, but the soil profile domain is arbitrarily heterogeneous. We employ the finite element method (FEM) to discretize the near-field soil. The semi-infinite far-field is represented using proper WABCs—namely, Perfectly Matched Layers (PML), which, to date, are considered to provide the best WABC regardless of the incident angles and frequencies of outgoing propagating waves [Kucukcoban (2011)]. A time domain wave solver is developed to compute the reaction forces for a given displacement time history. Then, the impedance matrix is computed for the frequency spectrum included in the loading time history. The method is verified against known solutions, and the effects of material heterogeneity on the impedance functions are explored. 1. THE NUMERICAL MODELING. Embedded Rigid Foundation Rigid Boundary 2 1
Finite Element Mesh Figure 3. A rigid embedded foundation in a 2D setting (the grid indicates the discretized finite element meshes and the dots indicate the corresponding nodes on the excavated foundation boundary). For a rigid foundation in a 2D setting (Fig. 2), the motion of the foundation can be described by three representative degrees of freedom—i.e., the vertical, horizontal, and rotational motions, which are denoted here as ∆1, ∆2, and θ, respectively. The displacement-reaction relationship of the rigid foundation in the frequency domain can be described by using the non-dimensionalized form of an impedance matrix as in
3
! Rˆ $ !KV # ˆ2 & # # R1 & = 'µ # 0 ˆ /b & #" M #" 0 %
0 $! (ˆ 2 $ &# & (Eq. 1) K HM &# (ˆ 1 & , K MM &%#"b)ˆ &% where, “^” denotes the Fourier-transform of the subtended variable. The parameter b ˆ and !ˆ , denotes the width of the foundation, and is used to normalize the variables M respectively, which denote the moment and the angle of rotation with respect to the foundation’s centroid. In what follows, we present a numerical procedure for computing the impedance matrix, shown in (Eq.1), for various soil profiles and different shapes of the foundation boundary (i.e., the interface between the near-field soil and the adjacent soil). The numerical procedure entails the following steps: (1) we compute the vertical reaction Rˆ 2 for a given vertical motion ( !ˆ 1 = 0 , !ˆ 2 " 0 , and !ˆ = 0 ) and ˆ for a given horizontal then compute KV; (2) we obtain the reactions Rˆ1 and M motion ( !ˆ 1 " 0 , !ˆ 2 = 0 , and !ˆ = 0 ) and then compute KHH and KMH; and finally, (3) ˆ for a given rotational motion ( !ˆ = 0 , !ˆ = 0 , we compute the reactions Rˆ1 and M 1 2 and !ˆ " 0 ) and then compute KHM and KMM. To this end, for given displacement conditions ( !1 (t) , ! 2 (t) , and ! (t) ), we compute reaction forces and moment (R1(t), R2(t), and M(t)) in the time domain. The displacements, as well as the reaction forces and the moment are, in turn, transformed from the time domain into the frequency domain via FFT. Then, we compute the impedance functions for the frequency spectrum included in the loading time history. Here, in order to capture the impedance matrix for a broad range of frequencies, we use a Ricker pulse, shown in Fig. 3, to modulate the rigid vertical, horizontal, and rotational movements ( !1 (t) , ! 2 (t) , and ! (t) ). 0 K HH K MH
ï3
1
ï6
10
x 10
x 10
0.8
5
u 2 (t )
|u ˆ 2(ω )|
0.6
0.4
0
0.2
0 ï5
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
Ti me t [ s]
0.7
0.8
0.9
1
(b)
0
2
4
6
8
10
12
Frequ en cy [ H z]
14
16
18
20
Figure 3. (a) The Ricker pulse loading time-history of the known displacement u2(t) with (b) the frequency spectrum of the central frequency fr = 5 Hz. To compute the time-domain wave responses, we consider a solid medium in a plane-strain setting as shown in Fig. 4. The regular domain ( !0 ) is truncated by PMLs. In the regular domain !0 , the wave motion is governed by the elastic wave equation. In the PML domain ( ! \ !0 ), a hybrid-PML formulation, developed by Kuckoban and Kallivokas (2011), gives rise to the attenuation of the amplitudes of both propagating and evanescent waves. The two wave equations are coupled with the proper interface conditions—viz., displacement and traction continuity conditions.
4
Known displacement x2
n x3
Γload
Γfree
ΩPML
ΩPML
Ω0
n
Γfixed ΩPML Γfixed n
Ω :The entire domain
Figure 4. The two-dimensional wave model for a semi-infinite soil medium. The wave response solutions are obtained via the standard finite element method, which is implemented on a high-performance-computing platform. The computed wave responses and impedance functions are rigorously verified by comparing them with reference solutions. Further formulation details and the verification studies are omitted here for brevity. 3. NUMERICAL EXPERIMENTS In this section, we discuss the variation of the impedance functions of rigid massless foundations for three different configurations (surface, embedded shape, and rectangular void), shown in Fig. 5, with respect to variations in the soil profile (here, we consider only horizontally-layered or linearly-depth-dependent soils). Massless Surface Strip Foundation
n
Massless Embedded Strip Foundation
n
Γfree
D
Γfree
D
Γfree
B
B
n
n
Ωreg
Ωreg
ΩPML
(a)
Ωreg
ΩPML
ΩPML
ΩPML
ΩPML
H
ΩPML B
ΩPML
(b)
ΩPML
(c)
ΩPML
Figure 5. Rigid foundations of different types in the 2D setting: (a) The strip surface foundation, (b) The embedded foundation, and (c) The rectangular void (tunnel). Numerical results: surface foundation resting on two-layered soil media Here, we consider two-layered soil media. The Young’s modulus of the top layer is E1 = 109 N/m2 and that of the underlying half-space is E2 = 1.5, 2, or 3×109 N/m2. The interface between the two layers is located at x2 = −30 m. The Poisson’s ratio and the mass density are ν = 0.25 and ρ = 2200 kg/m3 for both the top layer and the underlying half-space. We use a Ricker pulse with central frequency fr = 5 Hz.
5
1
1.4
Homog. soil
E1/E2 = 1/1.5
ˆ H H) I m( K
ˆ H H) R e( K
E1/E2 = 1/2 E1/E2 = 1/3
0.6
Homog. soil
1.2
E1/E2 = 1/1.5
0.8
0.4
1
E1/E2 = 1/2
0.8
E1/E2 = 1/3
0.6 0.4
0.2
0
0.2 0 0
0.5
1
1.5
2
Di men si on l ess frequ en cy a 0 =
0.5
1
1.5
2 ωB Vs
0.7
1
Homog. soil
0.6
E1/E2 = 1/1.5 ˆ M M) I m( K
0.8
ˆ M M) R e( K
0
Di men si on l ess frequ en cy a 0 =
ωB Vs
0.6
Homog. soil 0.4
E1/E2 = 1/1.5 E1/E2 = 1/2
0.5
E1/E2 = 1/2
0.4
E1/E2 = 1/3
0.3 0.2
E1/E2 = 1/3
0.2
0.1 0
0
0.5
1
1.5
Di men si on l ess frequ en cy a 0 =
2 ωB Vs
0
0
0.5
1
1.5
D i men si on l ess frequ en cy a 0 =
2 ωB Vs
Figure 6. The impedance functions (KHH and KMM ) of the strip surface foundation on the two-layered soils: VS denotes the shear velocity corresponding to the top layer soil, i.e., E1 = 109 N/m2, ν = 0.25, and ρ = 2200 kg/m3. As seen from Fig. 6, the impedance functions become more oscillatory as the stiffness contrast becomes more pronounced. The waviness occurs because of wave reflections between the ground surface and the layers’ interface. That is, for some frequencies, constructive interference occurs [Jeong et al. (2010), and Dobry et al. (1976)] which amplifies the reaction response at the foundation boundary (i.e., the real part of K increases at the given frequency), whereas for some other frequencies the interference is destructive. Numerical results: rectangular void (tunnel) foundation embedded in soil with linearly-depth-dependent stiffness A soil profile of a linearly-depth-dependent Young’s modulus is considered as shown in Fig. 7, where Ei denotes the Young’s modulus at x2(i), which is the depth of the i-th point of inflection of the gradient of the linearly-varying modulus. In this numerical experiment, we used E1 = 109 N/m2, and E2 = E3 = 0.5, 2, 3, or 4 ×109 N/m2; x2(1) is 0 m (the top ground surface); x2(2) is −80 m; and x2(3) is −120m. The Poisson’s ratio and the mass density are uniformly ν = 0.25 and ρ = 2200 kg/m3 for the entire soil domain. We consider a rectangular void with a half-width of B = 15 m and a half height of H = 15 m (cf. Fig. 5). The depth from the ground surface to the centroid of the void is D = 45 m. The location of the rotational centroid is (x1c, x2c) = (0 m, −45 m). The Ricker pulse loadings with central frequency fr = 5 Hz are used. Fig. 8 indicates that
6
both the real and the imaginary parts of KV, KHH, KMM, and KHM for this rectangular void foundation are highly sensitive to the gradient of the soil’s Young’s modulus. 0
E ( 1) at x 2( 1)
ï10 ï20 ï30
x 2 [ m]
ï40
E ( 2) at x 2( 2) ï50 ï60 ï70 ï80 ï90 ï100
E ( 3) at x 2( 3) 0
0.5
1
1.5
2
2.5
E (x 2 ) [ N/m2 ]
3 9
x 10
Figure 7. Linearly-depth-dependent Young’s modulus E(x2). 5
12
Homog. soil E1/E2 = 1/0.5
Homog. soil E1/E2 = 1/0.5
10
E1/E2 = 1/2
E1/E2 = 1/2
E1/E2 = 1/3 3
I m(K H H )
R e(K H H )
4
E1/E2 = 1/4
2
8
E1/E2 = 1/3 E1/E2 = 1/4
6 4
1
0
2 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Di men si on l ess frequ en cy a 0 =
1.6
1.8
2
E1/E2 = 1/2 E1/E2 = 1/4
I m(K M M )
R e(K M M )
E1/E2 = 1/3
10 8 6 4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.8
2
ωB Vs
7
Homog. soil E1/E2 = 1/0.5
6
E1/E2 = 1/2
5
E1/E2 = 1/3 E1/E2 = 1/4
4 3 2
2 0
0.4
8
Homog. soil E1/E2 = 1/0.5
12
0.2
D i m en si on l ess frequ en cy a 0 =
16 14
0
ωB Vs
1 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Di men si on l ess frequ en cy a 0 =
1.6
1.8
0
2
ωB Vs
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Di men si on l ess frequ en cy a 0 =
1.6 ωB Vs
Figure 8. The impedance functions (KHH and KMM) of the rectangular void foundation within soil of linearly-varying soil property: VS denotes the shear velocity corresponding to E1 = 109 N/m2, ν = 0.25, and ρ = 2200 kg/m3. 5. DISCUSSIONS We presented a numerical approach to computing impedance functions for rigid foundations with arbitrary shapes that are resting on (or embedded in) heterogeneous soil media. Perfectly Matched Layers are used for truncating the semi-infinite domain. The results indicate that the impedance functions are quite sensitive to the foundation type and geometry, as well as soil heterogeneity. This result suggests that impedance functions should be systematically investigated for various soil profiles and
7
foundation types so that they can be scaled and utilized for the seismic design of structures for which SSI is important. In future work, we plan to extend our method to the computation of impedance functions for flexible foundations, wherein an impedance function matrix that corresponds to all of the degrees of freedom on the flexible foundation boundary is computed. Alternatively, a reduced set of vibrational modes of the flexible foundation boundary can be selected and impedance functions can be computed for each of those modes. This would potentially result in large computational savings. REFERENCES Bielak, J., and Christiano, P. (1985). “On the effective seismic input for non-linear soil-structure-interaction systems.” Earthquake Engineering and Structural Dynamics, 12, 107–119. Dobry, R., Oweis, I., and Urzua, A. (1976). “Simplified procedures for estimating the fundamental period of a soil profile.” Computer Methods in Applied Mechanics and Engineering, 192, 1337–1375. Gazetas, G. (1984) “Seismic response of end-bearing single piles,” Soil Dynamics and Earthquake Engineering, 3(2), 82-93. Jeong, C., Kallivokas, L. F., Huh, C., and Lake, L.W. (2010). “Optimization of sources for focusing wave energy in targeted formations.” Journal of Geophysics and Engineering, 7, 242–256. Kucukcoban, S. and Kallivokas, L. F. (2011). “Mixed perfectly-matched-layers for direct transient analysis in 2D elastic heterogeneous media.” Computer Methods in Applied Mechanics and Engineering, 200, 57–76. Mylonakis, G., Nikolaou, A., and Gazetas, G. (1997). “Soil-pile-bridge seismic interaction: kinematic and inertial effects. Part I: s oft soil.” Earthquake Engineering and Structural Dynamics, 26, 337–359. Safak, E. (2006). “Time-domain representation of frequency-dependent foundation impedance functions.” Soil Dynamics and Earthquake Engineering, 26(1), 6570. Wolf, J. P. (1985). “Dynamic soil-structure interaction.” Prentice Hall. Wolf, J. P. (1989). “Soil-structure-interaction analysis in time domain.” Nuclear Engineering and Design, 111, 381–393.
8
