# Soil–Structure Interaction Consideration for Base Isolated Structures under Earthquake Excitation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

#### 2.1. Representing Soil Structure

_{s}and k

_{r}are swaying and rocking stiffness of the soil, respectively. Similarly, c

_{s}and c

_{r}are swaying and rocking and damping the soil, respectively. V

_{s}, $\rho $ and ν are the shear wave velocity, mass density and Poisson’s ratio of the soil. G is the shear modulus of the soil, which can be calculated with the following equation:

#### 2.2. Matrix Formulation of the Structure

**M**), damping (

**C**) and stiffness (

**K**) matrices for a fixed-based shear building can be written as:

_{1}to k

_{n}are the stiffnesses of the stories, c

_{1}to c

_{n}are the damping of the stories, and m

_{1}to m

_{n}are the mass of each story of an n-story structure. In the next section, the formulation for the base-isolated structure is presented.

#### 2.3. Base Isolated Structure

**M**), damping (

_{BI}**C**) and stiffness (

_{BI}**K**) matrices of a base-isolated n-story shear building can be written as:

_{BI}_{BI}is the stiffness, c

_{BI}is the damping and m

_{BI}is the mass of the BI system. BI adds one degree of freedom to the fixed base structure; therefore, the dimensions of all matrices are increased by one. SSI implementation to the expressions given above is defined in the next section.

#### 2.4. SSI Implementation

_{b}and I

_{b}are the mass and the mass moment of inertia of the foundation, respectively. I

_{1}to I

_{n}are the mass moment of inertia of each floor. k

_{s}is the swaying stiffness, k

_{r}is the rocking stiffness, c

_{s}is the swaying damping and c

_{r}is the rocking damping of the foundation-soil medium. h

_{1}to h

_{n}are story heights with respect to the ground. Swaying values are the horizontal components of the foundation–soil medium, whereas rocking values are rotational components. Mass, damping and stiffness matrices with SSI implementation can be written as follows [43]:

**M**is the mass matrix,

_{SSI}**C**is the damping matrix and

_{SSI}**K**is the stiffness matrix of the shear building considering SSI. In addition,

_{SSI}**M**and

_{v}**M**are defined as follows:

_{h}#### 2.5. Proposed Simple Formulation for SSI with Base Isolation

**M**,

**C**and

**K**matrices that are used in SSI matrices can be substituted with base-isolated

**M**,

_{BI}**C**and

_{BI}**K**matrices. The height of the BI system, h

_{BI}_{BI}, must be included in the mass matrix considering SSI and BI together.

**M**is the mass matrix,

_{SSI,BI}**C**is the damping matrix and

_{SSI,BI}**K**is the stiffness matrix of the shear building considering SSI and BI together.

_{SSI,BI}**M**and

_{v,BI}**M**can be presented as:

_{h,BI}#### 2.6. Equations of the Motion of the Structure under Earthquake Excitation

**m*** is a vector containing the diagonal of

**M**, which happens to be the mass value

**M**:

_{BI}**M**with $\left({m}_{b}+{{\displaystyle \sum}}_{i=1}^{n}{m}_{i}\right)$ and $\left({{\displaystyle \sum}}_{i=1}^{n}{m}_{i}{h}_{i}\right)$ from

**M**[43]:

_{SSI}**M**[43]:

_{SSI,BI}## 3. Numerical Example

_{b}, stiffness k

_{b}and damping c

_{b}are calculated with the following expressions. In the equations below, m is the mass, and k is the stiffness of the first story above the isolation level. m, k and c values are given in Table 4, Table 5 and Table 6. The following equation is taken from [1]. The critical damping ratio for base isolation is taken as 10%.

## 4. Numerical Results

#### Spectral Analysis

## 5. Conclusions

- BI may greatly reduce the acceleration, velocity and displacements of structures that are induced by earthquakes.
- Although the total displacement might be higher than in a fixed structure, most of it occurs on the isolation system, and the superstructure moves as a whole, resulting in much lower inter-story drift compared to the fixed base structure.
- SSI may modify the acceleration, velocity and displacement responses of structures.
- The results show that SSI mostly reduces the effects of earthquakes. In order to stay on the safer side, design codes do not specify SSI analysis procedures; they often briefly state that SSI can sometimes modify earthquake responses, and for those rare cases, it should be investigated.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Soil | ν | ρ ton/m ^{3} | V_{S}m/s ^{2} | G kN/m ^{2} |
---|---|---|---|---|

Very Soft | 0.49 | 1.60 | 50 | 4000 |

Soft | 0.49 | 1.80 | 100 | 18,000 |

Medium | 0.48 | 1.90 | 300 | 171,000 |

Dense | 0.33 | 2.40 | 500 | 600,000 |

Soil | k_{s}kN/m | k_{r}kN/m | c_{s}kNs/m | c_{r}kNs/m |
---|---|---|---|---|

Very Soft | 2.39 × 10^{5} | 3.00 × 10^{7} | 3.49 × 10^{5} | 1.01 × 10^{6} |

Soft | 1.07 × 10^{6} | 1.35 × 10^{8} | 7.87 × 10^{5} | 2.28 × 10^{6} |

Medium | 1.02 × 10^{7} | 1.26 × 10^{9} | 2.47 × 10^{6} | 7.09 × 10^{6} |

Dense | 3.24 × 10^{7} | 3.43 × 10^{9} | 4.74 × 10^{6} | 1.16 × 10^{7} |

Soil | k_{s}kN/m | k_{r}kN/m | c_{s}kNs/m | c_{r}kNs/m |
---|---|---|---|---|

Very Soft | 4.78 × 10^{5} | 2.40 × 10^{8} | 2.79 × 10^{6} | 1.62 × 10^{7} |

Soft | 2.15 × 10^{6} | 1.08 × 10^{9} | 6.30 × 10^{6} | 3.66 × 10^{7} |

Medium | 2.03 × 10^{7} | 1.01 × 10^{10} | 1.98 × 10^{7} | 1.14 × 10^{8} |

Dense | 6.48 × 10^{7} | 2.74 × 10^{10} | 3.79 × 10^{7} | 1.86 × 10^{8} |

story height, h_{i} (m) | 4 to 160 |

story mass, m_{i} (ton) | 980 |

story stiffness, k_{i} (kN/m) | 2.1 × 10^{6} to 0.99 × 10^{6} |

story damping, c_{i} (kNs/m) | 42.6 × 10^{3} to 20 × 10^{3} |

story inertia, I_{i} (ton·m^{2}) | 1.31 × 10^{5} |

foundation mass, m_{0} (ton) | 1960 |

foundation inertia, I_{0} (ton·m^{2}) | 1.96 × 10^{5} |

story height, h_{i} (m) | 4 to 20 |

story mass, m_{i} (ton) | 300 |

story stiffness, k_{i} (kN/m) | 3.5 × 10^{5} to 1.50 × 10^{5} |

story inertia, I_{i} (ton·m^{2}) | 7.5 × 10^{3} |

foundation mass, m_{0} (ton) | 300 |

foundation inertia, I_{0} (ton·m^{2}) | 7.5 × 10^{5} |

story height, h_{i} (m) | 4 to 40 |

story mass, m_{i} (ton) | 300 |

story stiffness, k_{i} (kN/m) | 7.0 × 10^{5} to 3.00 × 10^{5} |

story inertia, I_{i} (ton·m^{2}) | 7.5 × 10^{3} |

foundation mass, m_{0} (ton) | 300 |

foundation inertia, I_{0} (ton·m^{2}) | 7.5 × 10^{5} |

Event | Year | Magnitude (M_{w}) | Mechanism |
---|---|---|---|

Imperial Valley, US | 1940 | 6.95 | Strike Slip |

Imperial Valley, US | 1979 | 6.53 | Strike Slip |

Loma Prieta, US | 1989 | 6.93 | Reverse Oblique |

Manjil, IR | 1990 | 7.37 | Strike Slip |

Cape Mendocino, US | 1992 | 7.01 | Reverse |

Erzincan, TR | 1992 | 6.69 | Strike Slip |

Northridge, US | 1994 | 6.69 | Reverse |

Dinar, TR | 1995 | 6.40 | Normal |

Kobe, JP | 1995 | 6.90 | Strike Slip |

Chi-Chi, TW | 1999 | 7.62 | Reverse Oblique |

Düzce, TR | 1999 | 7.14 | Strike Slip |

Hector Mine, US | 1999 | 7.13 | Strike Slip |

Kocaeli, TR | 1999 | 7.51 | Strike Slip |

Iwate, JP | 2008 | 6.90 | Reverse |

Darfield, NZ | 2010 | 7.00 | Strike Slip |

El Mayor Cucapah, MX | 2010 | 7.20 | Strike Slip |

Christchurch, NZ | 2011 | 6.20 | Reverse Oblique |

Earthquake | Station | Soil | Fixed (m) | BI (m) | BI Reduction (%) | SSI (m) | SSI_BI (m) | BI Reduction w/SSI (%) |
---|---|---|---|---|---|---|---|---|

Imperial Valley, 1940 | El Centro Array #9 | Very Soft | 0.185 | 0.078 | 57.86 | 0.075 | 0.053 | 29.29 |

Imperial Valley, 1940 | El Centro Array #9 | Soft | 0.185 | 0.078 | 57.86 | 0.101 | 0.078 | 23.07 |

Imperial Valley, 1940 | El Centro Array #9 | Medium | 0.185 | 0.078 | 57.86 | 0.163 | 0.078 | 52.06 |

Imperial Valley, 1940 | El Centro Array #9 | Dense | 0.185 | 0.078 | 57.86 | 0.179 | 0.078 | 56.26 |

Imperial Valley, 1979 | Calipatria Fire Station | Very Soft | 0.070 | 0.022 | 68.48 | 0.022 | 0.016 | 26.70 |

Imperial Valley, 1979 | Calipatria Fire Station | Soft | 0.070 | 0.022 | 68.48 | 0.028 | 0.019 | 32.25 |

Imperial Valley, 1979 | Calipatria Fire Station | Medium | 0.070 | 0.022 | 68.48 | 0.057 | 0.022 | 61.18 |

Imperial Valley, 1979 | Calipatria Fire Station | Dense | 0.070 | 0.022 | 68.48 | 0.069 | 0.022 | 68.11 |

Imperial Valley, 1979 | Delta | Very Soft | 0.103 | 0.051 | 50.32 | 0.053 | 0.046 | 12.30 |

Imperial Valley, 1979 | Delta | Soft | 0.103 | 0.051 | 50.32 | 0.063 | 0.044 | 29.62 |

Imperial Valley, 1979 | Delta | Medium | 0.103 | 0.051 | 50.32 | 0.084 | 0.051 | 39.14 |

Imperial Valley, 1979 | Delta | Dense | 0.103 | 0.051 | 50.32 | 0.093 | 0.051 | 44.87 |

Loma Prieta, 1989 | Alameda Naval Air Stn Hanger | Very Soft | 0.195 | 0.036 | 81.59 | 0.026 | 0.030 | −14.31 |

Loma Prieta, 1989 | Alameda Naval Air Stn Hanger | Soft | 0.195 | 0.036 | 81.59 | 0.067 | 0.038 | 43.62 |

Loma Prieta, 1989 | Alameda Naval Air Stn Hanger | Medium | 0.195 | 0.036 | 81.59 | 0.162 | 0.036 | 77.66 |

Loma Prieta, 1989 | Alameda Naval Air Stn Hanger | Dense | 0.195 | 0.036 | 81.59 | 0.185 | 0.036 | 80.52 |

Loma Prieta, 1989 | Capitola | Very Soft | 0.261 | 0.057 | 78.07 | 0.102 | 0.043 | 57.31 |

Loma Prieta, 1989 | Capitola | Soft | 0.261 | 0.057 | 78.07 | 0.235 | 0.060 | 74.32 |

Loma Prieta, 1989 | Capitola | Medium | 0.261 | 0.057 | 78.07 | 0.238 | 0.059 | 75.32 |

Loma Prieta, 1989 | Capitola | Dense | 0.261 | 0.057 | 78.07 | 0.256 | 0.058 | 77.38 |

Erzincan, 1992 (EW) | Erzincan | Very Soft | 0.516 | 0.250 | 51.55 | 0.232 | 0.197 | 15.14 |

Erzincan, 1992 (EW) | Erzincan | Soft | 0.516 | 0.250 | 51.55 | 0.374 | 0.244 | 34.67 |

Erzincan, 1992 (EW) | Erzincan | Medium | 0.516 | 0.250 | 51.55 | 0.481 | 0.250 | 47.90 |

Erzincan, 1992 (EW) | Erzincan | Dense | 0.516 | 0.250 | 51.55 | 0.494 | 0.250 | 49.31 |

Erzincan, 1992 (NS) | Erzincan | Very Soft | 0.571 | 0.249 | 56.40 | 0.231 | 0.196 | 15.26 |

Erzincan, 1992 (NS) | Erzincan | Soft | 0.571 | 0.249 | 56.40 | 0.372 | 0.243 | 34.74 |

Erzincan, 1992 (NS) | Erzincan | Medium | 0.571 | 0.249 | 56.40 | 0.479 | 0.249 | 47.95 |

Erzincan, 1992 (NS) | Erzincan | Dense | 0.571 | 0.249 | 56.40 | 0.533 | 0.249 | 53.29 |

Cape Mendocino, 1992 | Cape Mendocino | Very Soft | 0.285 | 0.106 | 62.85 | 0.126 | 0.111 | 12.50 |

Cape Mendocino, 1992 | Cape Mendocino | Soft | 0.285 | 0.106 | 62.85 | 0.194 | 0.108 | 44.37 |

Cape Mendocino, 1992 | Cape Mendocino | Medium | 0.285 | 0.106 | 62.85 | 0.256 | 0.106 | 58.53 |

Cape Mendocino, 1992 | Cape Mendocino | Dense | 0.285 | 0.106 | 62.85 | 0.276 | 0.106 | 61.57 |

Cape Mendocino, 1992 | Shelter Cove Airport | Very Soft | 0.009 | 0.004 | 52.19 | 0.006 | 0.003 | 50.17 |

Cape Mendocino, 1992 | Shelter Cove Airport | Soft | 0.009 | 0.004 | 52.19 | 0.006 | 0.003 | 41.20 |

Cape Mendocino, 1992 | Shelter Cove Airport | Medium | 0.009 | 0.004 | 52.19 | 0.008 | 0.004 | 44.65 |

Cape Mendocino, 1992 | Shelter Cove Airport | Dense | 0.009 | 0.004 | 52.19 | 0.009 | 0.004 | 50.99 |

Northridge, 1994 | Anacapa Island | Very Soft | 0.017 | 0.002 | 85.23 | 0.005 | 0.002 | 59.54 |

Northridge, 1994 | Anacapa Island | Soft | 0.017 | 0.002 | 85.23 | 0.007 | 0.002 | 71.38 |

Northridge, 1994 | Anacapa Island | Medium | 0.017 | 0.002 | 85.23 | 0.014 | 0.002 | 83.04 |

Northridge, 1994 | Anacapa Island | Dense | 0.017 | 0.002 | 85.23 | 0.016 | 0.002 | 84.52 |

Northridge, 1994 | Canoga Park | Very Soft | 0.637 | 0.320 | 49.77 | 0.297 | 0.252 | 15.26 |

Northridge, 1994 | Canoga Park | Soft | 0.637 | 0.320 | 49.77 | 0.478 | 0.312 | 34.73 |

Northridge, 1994 | Canoga Park | Medium | 0.637 | 0.320 | 49.77 | 0.614 | 0.320 | 47.87 |

Northridge, 1994 | Canoga Park | Dense | 0.637 | 0.320 | 49.77 | 0.631 | 0.320 | 49.24 |

Kobe, 1995 | Abeno | Very Soft | 0.107 | 0.025 | 76.62 | 0.034 | 0.016 | 53.33 |

Kobe, 1995 | Abeno | Soft | 0.107 | 0.025 | 76.62 | 0.058 | 0.024 | 58.10 |

Kobe, 1995 | Abeno | Medium | 0.107 | 0.025 | 76.62 | 0.080 | 0.025 | 68.67 |

Kobe, 1995 | Abeno | Dense | 0.107 | 0.025 | 76.62 | 0.097 | 0.025 | 74.17 |

Kobe, 1995 | HIK | Very Soft | 0.103 | 0.029 | 71.54 | 0.032 | 0.016 | 48.03 |

Kobe, 1995 | HIK | Soft | 0.103 | 0.029 | 71.54 | 0.076 | 0.022 | 71.56 |

Kobe, 1995 | HIK | Medium | 0.103 | 0.029 | 71.54 | 0.092 | 0.029 | 68.64 |

Kobe, 1995 | HIK | Dense | 0.103 | 0.029 | 71.54 | 0.099 | 0.029 | 70.56 |

Dinar, 1995 | Balikesir | Very Soft | 0.005 | 0.002 | 60.24 | 0.003 | 0.001 | 52.65 |

Dinar, 1995 | Balikesir | Soft | 0.005 | 0.002 | 60.24 | 0.007 | 0.002 | 74.38 |

Dinar, 1995 | Balikesir | Medium | 0.005 | 0.002 | 60.24 | 0.005 | 0.002 | 65.64 |

Dinar, 1995 | Balikesir | Dense | 0.005 | 0.002 | 60.24 | 0.005 | 0.002 | 62.85 |

Dinar, 1995 | Dinar | Very Soft | 0.309 | 0.110 | 64.31 | 0.156 | 0.061 | 61.02 |

Dinar, 1995 | Dinar | Soft | 0.309 | 0.110 | 64.31 | 0.225 | 0.103 | 54.45 |

Dinar, 1995 | Dinar | Medium | 0.309 | 0.110 | 64.31 | 0.269 | 0.110 | 59.31 |

Dinar, 1995 | Dinar | Dense | 0.309 | 0.110 | 64.31 | 0.306 | 0.110 | 64.02 |

Kocaeli, 1999 | Arcelik | Very Soft | 0.053 | 0.016 | 70.20 | 0.017 | 0.014 | 16.73 |

Kocaeli, 1999 | Arcelik | Soft | 0.053 | 0.016 | 70.20 | 0.034 | 0.016 | 53.71 |

Kocaeli, 1999 | Arcelik | Medium | 0.053 | 0.016 | 70.20 | 0.052 | 0.016 | 69.52 |

Kocaeli, 1999 | Arcelik | Dense | 0.053 | 0.016 | 70.20 | 0.053 | 0.016 | 70.43 |

Kocaeli, 1999 | Istanbul | Very Soft | 0.014 | 0.010 | 31.95 | 0.009 | 0.009 | 1.23 |

Kocaeli, 1999 | Istanbul | Soft | 0.014 | 0.010 | 31.95 | 0.016 | 0.010 | 40.94 |

Kocaeli, 1999 | Istanbul | Medium | 0.014 | 0.010 | 31.95 | 0.014 | 0.010 | 28.99 |

Kocaeli, 1999 | Istanbul | Dense | 0.014 | 0.010 | 31.95 | 0.013 | 0.010 | 25.07 |

Chi-Chi, 1999 | CHY006 | Very Soft | 0.362 | 0.073 | 79.85 | 0.087 | 0.060 | 31.67 |

Chi-Chi, 1999 | CHY006 | Soft | 0.362 | 0.073 | 79.85 | 0.183 | 0.071 | 61.13 |

Chi-Chi, 1999 | CHY006 | Medium | 0.362 | 0.073 | 79.85 | 0.344 | 0.073 | 78.76 |

Chi-Chi, 1999 | CHY006 | Dense | 0.362 | 0.073 | 79.85 | 0.357 | 0.073 | 79.57 |

Chi-Chi, 1999 | CHY101 | Very Soft | 0.216 | 0.080 | 62.68 | 0.092 | 0.103 | −11.95 |

Chi-Chi, 1999 | CHY101 | Soft | 0.216 | 0.080 | 62.68 | 0.103 | 0.094 | 8.96 |

Chi-Chi, 1999 | CHY101 | Medium | 0.216 | 0.080 | 62.68 | 0.173 | 0.082 | 52.78 |

Chi-Chi, 1999 | CHY101 | Dense | 0.216 | 0.080 | 62.68 | 0.197 | 0.081 | 58.97 |

Duzce, 1999 | Bursa | Very Soft | 0.848 | 0.433 | 48.95 | 0.401 | 0.341 | 15.03 |

Duzce, 1999 | Bursa | Soft | 0.848 | 0.433 | 48.95 | 0.643 | 0.422 | 34.35 |

Duzce, 1999 | Bursa | Medium | 0.848 | 0.433 | 48.95 | 0.820 | 0.433 | 47.22 |

Duzce, 1999 | Bursa | Dense | 0.848 | 0.433 | 48.95 | 0.840 | 0.433 | 48.45 |

Duzce, 1999 | Duzce | Very Soft | 0.632 | 0.316 | 49.95 | 0.295 | 0.249 | 15.42 |

Duzce, 1999 | Duzce | Soft | 0.632 | 0.316 | 49.95 | 0.474 | 0.309 | 34.88 |

Duzce, 1999 | Duzce | Medium | 0.632 | 0.316 | 49.95 | 0.610 | 0.317 | 48.03 |

Duzce, 1999 | Duzce | Dense | 0.632 | 0.316 | 49.95 | 0.626 | 0.317 | 49.42 |

Manjil, 1990 | Abbar | Very Soft | 0.110 | 0.079 | 28.09 | 0.061 | 0.060 | 1.50 |

Manjil, 1990 | Abbar | Soft | 0.110 | 0.079 | 28.09 | 0.078 | 0.087 | −10.94 |

Manjil, 1990 | Abbar | Medium | 0.110 | 0.079 | 28.09 | 0.092 | 0.082 | 10.35 |

Manjil, 1990 | Abbar | Dense | 0.110 | 0.079 | 28.09 | 0.099 | 0.081 | 18.94 |

Manjil, 1990 | Rudsar | Very Soft | 0.057 | 0.015 | 73.12 | 0.026 | 0.012 | 55.12 |

Manjil, 1990 | Rudsar | Soft | 0.057 | 0.015 | 73.12 | 0.053 | 0.015 | 71.25 |

Manjil, 1990 | Rudsar | Medium | 0.057 | 0.015 | 73.12 | 0.061 | 0.015 | 75.50 |

Manjil, 1990 | Rudsar | Dense | 0.057 | 0.015 | 73.12 | 0.058 | 0.015 | 73.63 |

Hector Mine, 1999 | Banning | Very Soft | 0.011 | 0.006 | 40.51 | 0.006 | 0.004 | 34.64 |

Hector Mine, 1999 | Banning | Soft | 0.011 | 0.006 | 40.51 | 0.006 | 0.006 | 8.82 |

Hector Mine, 1999 | Banning | Medium | 0.011 | 0.006 | 40.51 | 0.010 | 0.006 | 35.18 |

Hector Mine, 1999 | Banning | Dense | 0.011 | 0.006 | 40.51 | 0.010 | 0.006 | 38.88 |

Hector Mine, 1999 | Indio | Very Soft | 0.073 | 0.044 | 39.99 | 0.046 | 0.035 | 24.68 |

Hector Mine, 1999 | Indio | Soft | 0.073 | 0.044 | 39.99 | 0.082 | 0.046 | 43.52 |

Hector Mine, 1999 | Indio | Medium | 0.073 | 0.044 | 39.99 | 0.067 | 0.044 | 34.91 |

Hector Mine, 1999 | Indio | Dense | 0.073 | 0.044 | 39.99 | 0.071 | 0.044 | 38.34 |

Iwate, 2008 | AKT023 | Very Soft | 0.186 | 0.036 | 80.50 | 0.043 | 0.030 | 31.37 |

Iwate, 2008 | AKT023 | Soft | 0.186 | 0.036 | 80.50 | 0.085 | 0.035 | 58.73 |

Iwate, 2008 | AKT023 | Medium | 0.186 | 0.036 | 80.50 | 0.161 | 0.036 | 77.71 |

Iwate, 2008 | AKT023 | Dense | 0.186 | 0.036 | 80.50 | 0.179 | 0.036 | 79.83 |

Iwate, 2008 | IWT010 | Very Soft | 0.139 | 0.067 | 51.61 | 0.062 | 0.043 | 30.69 |

Iwate, 2008 | IWT010 | Soft | 0.139 | 0.067 | 51.61 | 0.080 | 0.065 | 18.65 |

Iwate, 2008 | IWT010 | Medium | 0.139 | 0.067 | 51.61 | 0.129 | 0.068 | 46.80 |

Iwate, 2008 | IWT010 | Dense | 0.139 | 0.067 | 51.61 | 0.137 | 0.068 | 50.50 |

El Mayor-Cucapah, 2010 | Chihuahua | Very Soft | 0.133 | 0.040 | 69.61 | 0.038 | 0.053 | −38.85 |

El Mayor-Cucapah, 2010 | Chihuahua | Soft | 0.133 | 0.040 | 69.61 | 0.134 | 0.046 | 65.74 |

El Mayor-Cucapah, 2010 | Chihuahua | Medium | 0.133 | 0.040 | 69.61 | 0.138 | 0.041 | 70.22 |

El Mayor-Cucapah, 2010 | Chihuahua | Dense | 0.133 | 0.040 | 69.61 | 0.142 | 0.041 | 71.50 |

El Mayor-Cucapah, 2010 | Michoacan de Ocampo | Very Soft | 0.332 | 0.113 | 65.99 | 0.107 | 0.095 | 11.68 |

El Mayor-Cucapah, 2010 | Michoacan de Ocampo | Soft | 0.332 | 0.113 | 65.99 | 0.184 | 0.117 | 36.35 |

El Mayor-Cucapah, 2010 | Michoacan de Ocampo | Medium | 0.332 | 0.113 | 65.99 | 0.276 | 0.114 | 58.63 |

El Mayor-Cucapah, 2010 | Michoacan de Ocampo | Dense | 0.332 | 0.113 | 65.99 | 0.315 | 0.113 | 63.95 |

Darfield, 2010 | Canterbury Aero Club | Very Soft | 0.056 | 0.037 | 35.09 | 0.045 | 0.029 | 34.56 |

Darfield, 2010 | Canterbury Aero Club | Soft | 0.056 | 0.037 | 35.09 | 0.052 | 0.039 | 24.88 |

Darfield, 2010 | Canterbury Aero Club | Medium | 0.056 | 0.037 | 35.09 | 0.050 | 0.037 | 25.14 |

Darfield, 2010 | Canterbury Aero Club | Dense | 0.056 | 0.037 | 35.09 | 0.053 | 0.037 | 30.06 |

Darfield, 2010 | DSLC | Very Soft | 0.218 | 0.064 | 70.61 | 0.053 | 0.046 | 12.41 |

Darfield, 2010 | DSLC | Soft | 0.218 | 0.064 | 70.61 | 0.071 | 0.071 | 0.30 |

Darfield, 2010 | DSLC | Medium | 0.218 | 0.064 | 70.61 | 0.196 | 0.066 | 66.20 |

Darfield, 2010 | DSLC | Dense | 0.218 | 0.064 | 70.61 | 0.215 | 0.065 | 69.82 |

Christchurch, 2011 | ADCS | Very Soft | 0.017 | 0.003 | 81.89 | 0.006 | 0.003 | 46.34 |

Christchurch, 2011 | ADCS | Soft | 0.017 | 0.003 | 81.89 | 0.006 | 0.003 | 49.92 |

Christchurch, 2011 | ADCS | Medium | 0.017 | 0.003 | 81.89 | 0.014 | 0.003 | 78.63 |

Christchurch, 2011 | ADCS | Dense | 0.017 | 0.003 | 81.89 | 0.016 | 0.003 | 80.98 |

Christchurch, 2011 | CECS | Very Soft | 0.009 | 0.006 | 38.67 | 0.009 | 0.003 | 64.57 |

Christchurch, 2011 | CECS | Soft | 0.009 | 0.006 | 38.67 | 0.007 | 0.004 | 31.37 |

Christchurch, 2011 | CECS | Medium | 0.009 | 0.006 | 38.67 | 0.008 | 0.006 | 30.16 |

Christchurch, 2011 | CECS | Dense | 0.009 | 0.006 | 38.67 | 0.009 | 0.006 | 36.30 |

Soil | Base | T (s) | $\mathbf{SA}(\mathbf{m}/{\mathbf{s}}^{2})$ | SV (m/s) | SD (m) |
---|---|---|---|---|---|

Very Soft Soil | fixed | 0.82 | 5.34 | 0.70 | 0.09 |

Very Soft Soil | BI | 2.13 | 1.93 | 0.65 | 0.22 |

Very Soft Soil | SSI | 1.12 | 3.44 | 0.61 | 0.11 |

Very Soft Soil | SSI+BI | 2.26 | 1.84 | 0.66 | 0.24 |

Soft Soil | fixed | 0.82 | 5.34 | 0.70 | 0.09 |

Soft Soil | BI | 2.13 | 1.93 | 0.65 | 0.22 |

Soft Soil | SSI | 0.89 | 4.82 | 0.68 | 0.10 |

Soft Soil | SSI+BI | 2.16 | 1.92 | 0.66 | 0.23 |

Medium Soil | fixed | 0.82 | 5.34 | 0.70 | 0.09 |

Medium Soil | BI | 2.13 | 1.93 | 0.65 | 0.22 |

Medium Soil | SSI | 0.83 | 5.45 | 0.72 | 0.09 |

Medium Soil | SSI+BI | 2.13 | 1.92 | 0.65 | 0.22 |

Dense Soil | fixed | 0.82 | 5.34 | 0.70 | 0.09 |

Dense Soil | BI | 2.13 | 1.93 | 0.65 | 0.22 |

Dense Soil | SSI | 0.82 | 5.39 | 0.70 | 0.09 |

Dense Soil | SSI+BI | 2.13 | 1.92 | 0.65 | 0.22 |

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## Share and Cite

**MDPI and ACS Style**

Yanik, A.; Ulus, Y.
Soil–Structure Interaction Consideration for Base Isolated Structures under Earthquake Excitation. *Buildings* **2023**, *13*, 915.
https://doi.org/10.3390/buildings13040915

**AMA Style**

Yanik A, Ulus Y.
Soil–Structure Interaction Consideration for Base Isolated Structures under Earthquake Excitation. *Buildings*. 2023; 13(4):915.
https://doi.org/10.3390/buildings13040915

**Chicago/Turabian Style**

Yanik, Arcan, and Yalcincan Ulus.
2023. "Soil–Structure Interaction Consideration for Base Isolated Structures under Earthquake Excitation" *Buildings* 13, no. 4: 915.
https://doi.org/10.3390/buildings13040915