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Edited by: Nikos D. Lagaros, National Technical University of Athens, Greece

Reviewed by: Vagelis Plevris, School of Pedagogical and Technological Education, Greece; Sameh Samir F. Mehanny, Cairo University, Egypt; Anaxagoras Elenas, Democritus University of Thrace, Greece

Specialty section: This article was submitted to Earthquake Engineering, a section of the journal Frontiers in Built Environment

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Although the energy transmitting boundary (TB) is accurate and efficient for the finite element method earthquake response analysis, it could be applied in the frequency domain only. In the previous papers, the author proposed an earthquake response analysis method using the time domain energy TB for two-dimensional (2D) problems. In this paper, this technique is expanded for three-dimensional (3D) problems. The inner field is supposed to be a hexahedron shape, and the approximate time domain boundary is explained, first. Next, 2D antiplane time domain boundary is studied for a part of the approximate 3D boundary method. Then, accuracy and efficiency of the proposed method are confirmed by example problems.

In order to accurately estimate the behavior of buildings during severe earthquakes, both the soil–structure interaction and non-linear effects must be taken into consideration. In addition, three-dimensional (3D) models are needed to express the complex shape of buildings, basements, and piles. In the case where buildings are built close to each other, structure–soil–structure interaction, i.e., Lou et al. (

Although the soil has a semi-infinite extent, the soil model needs to be generated as a finite region model in the FEM analyses. Therefore, artificial wave boundary models are needed especially at the side of the soil model. Currently, simple models, such as the cyclic boundary and the viscous boundary (VB) (Lysmer and Kuhlelameyer,

Many investigations into this problem have been conducted, i.e., Smith (

In contrast, the energy transmitting boundary (TB) used in FLUSH (Lysmer et al.,

The author has previously proposed the time domain transform methods of strongly frequency-dependent dynamic stiffness and proved that these methods are accurate yet simple (Nakamura,

In this paper, 3D time history FEM analyses with TB are studied based on these results. The axisymmetric boundary model used in ALUSH is known as a 3D problem TB. However, in many cases, the orthogonal coordinate system is preferred to the axisymmetric coordinate system for actual problems as shown in Figure

Shape of inner field | ||

Coordinate system | Axisymmetric | Orthogonal |

Transmitting boundary | Theoretical method | Approximate method (proposed method) |

Accordingly, the TB should also be formulated using orthogonal coordinates rather than axisymmetric coordinates, but it is not possible to obtain such a theoretical solution. Therefore, an approximate 3D boundary model (hereinafter referred to as 3D-TB model) from a combination of a 2D in-plane problem TB (hereinafter referred to as SV-TB) and a 2D antiplane problem TB (Lysmer and Waas,

At first, the outline of this model is explained. Next, a component of the model, the SH-TB, which has not been studied using time domain transform, is studied.

Then, the characteristics of soil impedance and input motion using 3D-TB are studied. Finally, time history response analysis of the 3D soil–structure interaction system using proposed 3D-TB model is conducted, and the effectiveness of the model is evaluated.

The VB, which is currently thought to be the most practical method for time domain analysis, is used for comparison in this study. Furthermore, it is known that accuracy is improved if the excavation force (EF) is applied to VB. EF is a correction force vector calculated as the product of free field soil displacements and frequency-independent stiffness matrix (refer to Supplementary Material). In order to further clarify the practical applicability of the proposed method, VB with EF is also compared in this study.

The TB is a highly accurate boundary model located at the outer side of the inner soil model, which is formed by parallel layers on the rigid bedrock. In a horizontal direction, the formulation is theoretical and rigorous. In a vertical direction, the formulation is approximate since it follows the element displacement assumption. The TB is able to almost completely absorb wave motion from an arbitrary direction. Even when the bottom of soil is semi-infinite condition, a favorable evaluation is possible by adding a sufficient amount of elements to the soil bottom, in the frequency domain.

In this paper, a time domain 3D-TB model, which corresponds to orthogonal coordinate system and uses SV-TB and SH-TB approximately, is proposed. An outline of this is described hereinafter.

The image of an inner field model is shown in Figure

The control width of one nodal line extends to the center of the adjacent nodal lines. Both SV-TB and SH-TB are assigned in this nodal line (refer to Figure

Furthermore, if the soil properties are the same, each nodal line becomes a TB with identical properties, and only the control width is different. For this reason, a TB with a unit width nodal line that corresponds to the type of the soil properties is prepared. This is multiplied by the control width and assigned into the entire boundary surface.

The analysis flow is shown in Figure

The reaction force from TB has to be calculated in the time domain. The calculation is not easy, because the components of the TB matrix are strongly frequency dependent. In this section, the concept of the transform of TB to the time domain and the obtained reaction force in the time domain are briefly explained, using a simple single DOF equation.

Although many methods to transform frequency dependent impedance function to the time domain have been proposed, most of them employed either the past displacement or the past velocity in the formulation of the impulse response. The author proposed transform methods using both the past displacement and velocity, then he confirmed that the accuracy of these methods is high (Nakamura,

In this paper, the following methods were used for the transform. Here, Eq.

_{j} = _{0}_{j}, _{1}_{j}, and _{2}_{0} are the coefficients of the impulse response. _{0}_{0}, _{1}_{0}, and _{2}_{0} are called the simultaneous components, because they correspond to the current time _{1}_{1} ~ _{1}_{n’} and _{0}_{1} ~ _{0}_{n’} are called the time-delay components, since they correspond to the past time (_{j}). All of the unknown coefficients of the impulse response are obtained by simultaneous equations with given function data for

In the case when the hysteretic damping is large, the accuracy of the transform tends to decrease. To improve this problem, the simultaneous components (_{2}_{0}, _{1}_{0}, and _{0}_{0}) are corrected with (Δ_{2}_{0}, Δ_{1}_{0}, and Δ_{0}_{0}), where Δ_{2}_{0}, Δ_{1}_{0}, and Δ_{0}_{0} indicate the modification terms determined by the least square method. The improved reaction force [

Using Eqs _{B}] can be transformed to the time domain. The details of the transform are shown in Nakamura (

With the method proposed in this paper, the nodal lines are mutually discontinuous as above. Therefore, it is thought that accuracy will decrease when the neighboring free field soil conditions differ greatly. Furthermore, it is necessary to calculate the SV-TB and SH-TB of the nodal line for each type of soil properties. Thus, when there are many types of soil properties, the calculation time increases, and the analysis becomes less efficient. For this reason, it is thought that the proposed method is effective when the types of soil properties are not so many.

At first, the properties and applicability of the time domain SH-TB, which is a component of the 3D-TB model, are verified in preparation for analysis of this model. The applicability and accuracy of SV-TB was already confirmed in Nakamura (

The analysis model is shown in Figure _{s} in the range 200–400 m/s, on the bedrock with _{s} = 500 m/s. A height difference of 10 m is set at one side of the soil (only left side). The characteristics of the bedrock are evaluated using the bottom VB in the inner field.

The building is represented by a lumped mass model with shear elements. Its width is 20 m, the height of above-ground part is 24 m, and the height of the underground part is 10 m. The causal hysteretic damping model (Nakamura,

_{s} (m/s) |
Poisson ratio |
Density ^{3}) |
Damping ratio |
Thickness (m) | |
---|---|---|---|---|---|

Surface 1 | 200 | 0.4 | 2.0 | 0.02 | 20 |

2 | 300 | 10 | |||

3 | 400 | 10 | |||

Bedrock | 500 | 0 | – |

Story | Height (m) | Weight (t) | Rotational inertia (^{5} t/m^{2}) |
Shear stiffness (^{6} kN/m) |
---|---|---|---|---|

6 | 4.0 | 480 | 0 | 0.4935 |

5 | 4.0 | 480 | 0 | 0.9047 |

4 | 4.0 | 480 | 0 | 1.234 |

3 | 4.0 | 480 | 0 | 1.480 |

2 | 4.0 | 480 | 0 | 1.645 |

1 | 4.0 | 480 | 0 | 1.727 |

B1 | 5.0 | 720 | 0 | ∞ |

B2 | 5.0 | 720 | 1.68 | ∞ |

Three analysis models, with the boundary at a distance of

For estimating the semi-infinity of the bottom soil, the elements for 100 m height of the material properties of the bedrock were added to the lowest part of the soil model in the calculation of the TB matrix. The conditions for time domain transform for the TB matrix are shown in Table

Impedance | Impulse response | |||
---|---|---|---|---|

No. of data | Frequencies of data (Hz) | Δ |
Simultaneous components | Time delay components |

21 | 0.1, 1.0, 2.0, 3.0, …, 19.0, 20.0 | 0.05 | _{0}, _{0}, _{0} |
_{1} ~ _{20}, _{1} ~ _{19} |

The input ground motion was El Centro 1940NS wave (duration of 10 s, time step Δ

Figure

In the study of the SV problem (Nakamura,

Incidentally, when input ground motion from a vertically downward direction is assumed for the SH problem, as it is in this analysis, EF is not required in the calculation.

The characteristics of soil impedance and input motion were evaluated in order to study the efficiency of the proposed 3D-TB. The same study was conducted with a VB as a target of comparison.

The massless rigid foundation embedded in the multilayered soil is studied. The FEM analysis model is shown in Figure

Impulse excitation was performed for the massless rigid foundation in order to calculate the time history wave for the displacement of the foundation. The impulse excitation consists of many different and constant frequency components (it is so-called as the white noise), so it is very convenient to study the frequency dependency of the given function in the time domain. The time integral method was the same as that described in the previous section. The impulse excitation time history wave and the foundation displacement time history wave were transformed using Fourier transformation, and the divisions for frequency domain were performed to calculate impedance. Two components of impedance – horizontal (_{x}) and rotational (_{θy}) – were studied. The thin-layer element method (TLEM) (Tajimi,

Figure

For the rotational component in Figure

An impulse wave was applied as the input ground motion from the bottom of the model, and time history response analysis is conducted. The acceleration response wave is calculated at the center of the massless rigid foundation at soil surface level. The acceleration response and the time history wave of the impulse input motion are transformed to the frequency domain by Fourier transform, and divisions are performed to calculate the transfer function of the input motion. Two studies were conducted for VB, one when EF is applied and one when EF is not considered.

The analysis results are shown in Figure

The accuracy of soil impedance and input motion was studied for the massless rigid foundation in the multilayered soil. The results when using 3D-TB were as follows.

For soil impedance, the results corresponded favorably with the analytical solution (TLEM) generally. Although there was slight fluttering in the case of

All cases are almost identical for input motion.

In contrast, the results when using VB were as follows.

For soil impedance, there was a large difference in the case of

In the case of input motion, the disparity in each case was large when EF was not applied, and in all cases, the results differed from the results for the TB. Accuracy improved when EF was applied, and the results corresponded with the results for the TB in all cases except for

Time history seismic analysis of the soil and structure interaction system is conducted using the proposed 3D-TB, and the accuracy and the efficiency of the method are studied.

The analysis model is shown in Figure

The maximum response values (acceleration and displacement) for soil near the building when 3D-TB is used are shown in Figure

The maximum response values using VB without EF are shown in Figure

The results for VB at

The results when VB with EF is used are shown in Figure

Figure

Table

Case | Excitation | Acceleration | Displacement | Shear force |
---|---|---|---|---|

0.93–1.01 | 0.95–1.01 | 0.97–1.01 | ||

0.90–0.95 | 0.94–0.99 | 0.91–0.95 | ||

0.99–1.01 | 0.99–1.01 | 0.99–1.01 | ||

0.96–0.99 | 0.97–1.00 | 0.97–0.97 | ||

0.73–0.91 | 0.91–1.06 | 0.75–0.84 | ||

0.73–0.89 | 0.97–1.07 | 0.84–0.88 | ||

0.82–0.91 | 0.87–0.99 | 0.82–0.84 | ||

0.79–0.92 | 0.87–0.98 | 0.79–0.84 | ||

0.89–0.92 | 0.91–0.99 | 0.90–0.92 | ||

0.89–0.91 | 0.91–0.98 | 0.88–0.90 | ||

0.95–0.99 | 0.95–0.99 | 0.96–0.99 | ||

0.96–0.98 | 0.96–0.98 | 0.96–0.97 | ||

0.71–0.90 | 0.82–0.94 | 0.73–0.82 | ||

0.74–0.93 | 0.87–0.96 | 0.81–0.86 | ||

0.83–0.96 | 0.89–0.96 | 0.86–0.93 | ||

0.83–0.96 | 0.93–0.98 | 0.92–0.95 | ||

0.94–0.98 | 0.94–0.99 | 0.95–0.98 | ||

0.95–0.98 | 0.95–0.99 | 0.96–0.98 | ||

0.98–0.99 | 0.99–0.99 | 0.98–0.99 | ||

0.96–1.00 | 0.98–0.99 | 0.96–0.98 |

The maximum response values for the above-ground part of the building when VB without EF is used are shown in Figure

Figure

However, the accuracy for

The above tendency is consistent with the results in Section “Soil Impedance and Input Motion of the 3D-TB.” Thus, the following can be concluded.

When the 3D-TB is used, response accuracy is favorable even when

When VB without EF is used, the accuracy of the response results is low at

When VB with EF is used, the response values at

Table

No. of node | No. of elem. | Required memory (GB) | Analysis time (min) | ||
---|---|---|---|---|---|

TB | 5 | 6,675 | 5,520 | 1.3 | 39 |

VB without EF | 80 (16.0) | 195,135 (29.2) | 184,336 (33.4) | 18.0 (13.8) | 516 (13.2) |

VB with EF | 20 (4.0) | 23,631 (3.5) | 21,136 (3.8) | 2.4 (1.8) | 71 (1.8) |

As for the analysis load, the required memory size and the analysis time during the calculation were counted using a single core Xeon7560 (2.26 GHz) processor. This processing unit has 256 GB of main memory space, and the calculations for all cases were conducted within the main memory. Furthermore, the 3D-TB calculation time in the frequency domain and the time domain transform time (total for both for the SV problem and the SH problem is 1.2 min) are included in the 3D-TB analysis time.

Compared to VB (without EF) case, the 3D-TB case has around 1/30 of the number of inner field nodal points and elements. It is also ~1/13 of the memory and analysis time. Furthermore, this required memory and analysis time is reduced to approximately half that required in the case of VB with EF applied.

In this paper, an approximate time domain TB that can be used with a 3D orthogonal coordinate system was studied. First, 3D-TB with high calculation efficiency that can be applied in a rectangular analysis domain was explained. A nodal line on the boundary surface is considered to be a single unit, and the SV-TB and the SH-TB are assigned to it.

Next, the properties of the component, the SH-TB, were studied and verified for favorable accuracy. Then the impedance and input motion of the rigid foundation embedded in the multilayered soil were calculated using the 3D-TB, and favorable correspondence with the analysis solution was obtained.

Furthermore, seismic response analysis of the 3D problem was conducted using the proposed 3D-TB. From the aspect of the accuracy of horizontal response values, improvement effects were obtained at ~1/13 of the required memory and analysis time compared to VB without EF and approximately half of the required memory and analysis time compared to VB with EF. It can be concluded from this that the effectiveness of the proposed 3D-TB has been confirmed.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at