# On the Seismic Design of Structures with Tilting Located within a Seismic Region

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### Buildings with Asymmetric Yielding Produced by Tilting in Mexico City

^{2}). It indicates that there are hundreds of tilted buildings in the soft soil of the valley of Mexico. Most of the tilted buildings that were identified in this zone have 3 to 6 stories; however, there are some that have 8 or more stories. It is important to mention that most of the buildings presented in Figure 3 are old constructions, and probably, they do not comply with the current seismic regulations. For example, some of these buildings were designed for a considerably lower lateral strength than the suggested in the current MCBC, i.e., the recommended design spectra ordinates are higher nowadays than several years ago. Therefore, the fact that this kind of buildings present tilting problems and that probably do not comply with the current seismic regulations may lead to catastrophic seismic structural performance if a high intensity ground motion occurs in that area. For these reasons, it is important to propose seismic strength amplification factors for structures exhibiting asymmetric yielding produced by tilting and other reasons with the aim of incorporating the factors in future reliability-based seismic regulations. It is worth mentioning that the methodology used in this study can be applied to structural systems with similar problems, located worldwide.

## 2. Simplified Structural Models

## 3. General Methodology

#### 3.1. Evaluation of the Structural Reliability

_{a}(T

_{1})) [42], and the average spectral acceleration over a range of vibration periods S

_{aavg}[43]. However, there are other approaches that have been proposed to estimate the structural reliability, which are not based on scalar quantities. Such approaches represent the IM as a vector [44,45], but they are out of the scope of this study.

_{EDP}(y), using the following equation [39,40]:

_{1}). In the present study, the spectral acceleration at the fundamental vibration period as a fraction of gravity (S

_{a}(g), T

_{1}) and the maximum ductility demand (µ) of the structures are selected as IM, and EDP, respectively.

#### 3.2. Steps to Follow for the Proposed Methodology

- (1)
- Firstly, several nonlinear time history analyses are carried out for simplified structural systems with different characteristics of base shear coefficient (c), asymmetric level (α), and vibration period (T
_{1}). This is performed with the aim of obtaining the EDP as a function of seismic intensity. The maximum ductility demand of the systems (µ) is taken here as the EDP of interest. In order to calculate the maximum ductility demand of the systems, it is necessary to estimate first the ductility demand in both E-W and N-S directions (µ_{x}, µ_{y}) considering the simultaneous action of both horizontal components of the seismic ground motions; µ_{x}and µ_{y}are calculated by taking the maximum horizontal displacement of the center of mass of the structural systems in each direction (d_{X}, d_{Y}) divided by their yield displacement, d_{y}, which in turn is estimated by a nonlinear static analysis). Finally, the maximum ductility demand is defined in this study as the maximum of the ductility values estimated in each horizontal direction, as indicated in Equation (2).$$\mathsf{\mu}=\mathrm{max}\left({\mathsf{\mu}}_{\mathrm{x}},{\mathsf{\mu}}_{\mathrm{y}}\right)$$ - (2)
- Next, the median (D) and standard deviation (σ
_{lnD}) of the ductility demand logarithms are calculated. - (3)
- Fragility curves for several values of the maximum ductility demand are obtained using Equation (3):$$\mathrm{P}\left(\mathrm{EDP}>\mathrm{y}|\mathrm{IM}=\mathrm{im}\right)=1-\mathsf{\Phi}\left(\frac{\mathrm{ln}\left(\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{D}$}\right.\right)}{{\mathsf{\sigma}}_{\mathrm{lnD}}}\right)$$
- (4)
- Ductility demand hazard curves (DDHC, Equation (1)) are obtained for symmetric yielding systems, and alternatively, for systems with different levels of asymmetric yielding.
- (5)
- Considering the ductility demand hazard curves corresponding to a wide variety of systems with different characteristics, ductility uniform exceedance rate spectra (µ-UERS) are obtained for several mean annual rate of exceedance values. To explicitly display the increment in the expected ductility demand of asymmetric yielding systems with respect to symmetric systems, ratios of µ-UERS corresponding to asymmetric yielding systems with respect to symmetric ones are calculated.
- (6)
- The next step is to obtain base shear coefficient spectra (BSCS) for symmetric as well as for asymmetric yielding systems employing a linear interpolation process. This procedure consists in selecting a value of the ductility demand and the associated values of T
_{1}and c, corresponding to a given µ-UERS. The process is repeated several times in order to obtain a data set T_{1}vs c for each value of the ductility demand considered. More details about this process can be found in [22]. Ratios between BSCS of systems with different levels of asymmetric yielding with respect to symmetric systems, R_{BSCS}, are calculated with the objective of quantifying the additional lateral strength requirement of systems with asymmetric yielding to achieve a seismic performance equivalent to their symmetric counterparts. These ratios can be expressed as:$${\mathrm{R}}_{\mathrm{BSCS}}=\frac{\mathrm{c}\left[\mathrm{BSCS}\left({\mathrm{T}}_{1},\mathsf{\nu},\mathsf{\mu},\mathsf{\alpha}\right)\right]}{\mathrm{c}\left[\mathrm{BSCS}\left({\mathrm{T}}_{1},\mathsf{\nu},\mathsf{\mu},\mathsf{\alpha}=0\right)\right]}$$ - (7)
- A simplified mathematical expression is fitted to the ratios of the base shear coefficient spectra obtained in step 6. The proposed expression is a function of the asymmetry level (α) of the structural system, the ratio between the fundamental vibration period of the system and the dominant period of the soil, and the maximum ductility demand of the system.
- (8)
- Steps (1) to (7) are repeated for the seismic zones of interest (having different soil dominant periods, from firm ground to very soft soil). The resulting mathematical expressions for each zone will be compared between them to evaluate the influence of the dominant period of the soil on the strength amplification factors. Hence, general rules for the seismic region under study can be proposed.

## 4. Seismic Zones Analyzed

_{s}(see Table 1). To address this concern, several seismic ground motions recorded in different stations were selected, their main characteristics can be consulted in [15]. The seismic records correspond to subduction events with moment magnitude greater than or equal to 6.9 (M ≥ 6.9), and approximately similar epicentral distances. It can be observed in Table 1 that the dominant period of the selected seismic zones varies from approximately 0.5 s for Zone A (firm ground) to approximately 3.5–4 s for Zone G (very soft soil).

_{aEW}and S

_{aNS}are the pseudo-acceleration elastic response spectra ordinates associated to the fundamental vibration period of the system under consideration, for 5% of critical damping, corresponding to E-W and N-S ground motions components, respectively. The seismic ground motions were scaled for S

_{a}/g values from 0.1 to 1.4. The scaling of the seismic records consists of multiplying the ordinates of the accelerograms using some factors in order to achieve that all the records present the same value of S

_{a}at the fundamental vibration period of the structure under consideration [42]. Figure 6 shows the pseudo-acceleration elastic response spectra for 5% of critical damping for the E-W component of the selected ground motions, along with their corresponding arithmetic average spectrum, which is represented by a bold black line.

## 5. Mathematical Expressions of Strength Amplification Factors Corresponding to the Seismic Region of Interest

_{BSCS}(defined in Equation (4)) for zones A to G corresponding to a ductility demand of 2 (μ = 2) and annual rate of exceedance ν = 0.008 (corresponding to a return period, T

_{r}= 125 years). The horizontal axis represents the ratio of the fundamental vibration period of the structural systems to that of the dominant period of the soil corresponding to the seismic zone under consideration; while the vertical axis shows the parameter R

_{BSCS}which represents the additional strength required for asymmetric yielding systems to achieve a seismic performance and structural reliability, in terms of the ductility demand, equivalent to their symmetric yielding counterparts.

_{BSCS}parameter is smaller for firm ground (Zones A and B) than for intermediate and soft soils (Zones C, D, E, F, and G) in the zone where the effect of asymmetric yielding is more important (i.e., T

_{1}/T

_{s}≈ 1). This indicates that the detrimental effect of asymmetric yielding is, generally, higher on soft soil than on firm ground; however, this is not always true, particularly for T

_{1}/T

_{s}ratios away from unity. In contrast, it is observed that for Zones A and B the shapes of the graphs are almost constant for T

_{1}/T

_{s}greater than 1. However, for the other seismic zones, the strength requirement reaches its maximum where the vibration period of the system is close to the dominant period of the soil and decreases as the T

_{1}/T

_{s}ratio moves away from unity. In summary, Figure 7 demonstrates that the effect of asymmetric yielding is more detrimental for structural systems located on soft soils and whose vibration period is close to the dominant period of the soil where is located.

_{1}) and the dominant period of the soil (T

_{s}). The general forms of the mathematical expressions proposed here are an extension of those proposed by Teran-Gilmore and Arroyo-Espinoza [21], which are based on a statistical analysis of the response of SDOF systems, under one-directional ground motions, where the SDOF systems are idealized with asymmetric force-deformation relationship (Figure 5), and in accordance with a constant damage criterion using the Park and Ang damage index [47]. On the other hand, the expressions proposed in this paper are based on a reliability analysis, which estimates mean annual rates of exceedance of an EDP for a given return period. In addition, the structural ductility demand and the level of asymmetric yielding are explicitly considered.

_{BSCS}ratios (as those in Figure 7) are:

_{BSCS}associated with a ductility demand of 2. It can be seen in Figure 8 that Equations (6) and (7) fit appropriately to the R

_{BSCS}data.

#### Comparison of the Proposed Mathematical Expressions with that Recommended in the Current Mexico City Building Code

_{1}/T

_{s}ratios considered. On the other hand, for intermediate and soft soils, the expression recommended by MCBC-2004 leads to conservative results only for T

_{1}/T

_{s}ratios away from unity; nevertheless, there is an important underestimation of the strength amplification factors for structures whose fundamental vibration period is close to the dominant period of the soil (i.e., T

_{1}/T

_{s}≈ 1). Results indicate that this underestimation may be higher than 100% (depending on the level of asymmetric yielding and on the ductility value). It is noticed that the MCBC-2004 recommends factors that are constant for all T

_{1}/T

_{s}values. This is an important limitation considering that the results presented in this study demonstrated that the effect of the effect of asymmetric yielding is more important for structural systems with vibration period close to the dominant period of the soil. The differences between the curves obtained in this study with respect to the MCBC-2004 are due to the fact that the former were obtained from a reliability-based analysis, while the latter were derived from a brief deterministic constant ductility criterion and engineering judgment.

## 6. Ductility Transformation Factors between Simplified and MDOF Systems

#### 6.1. Characteristics of the Buildings Analyzed

_{c}= 29.4 MPa for concrete, and f

_{y}= 411.9 MPa in tension and compression for the reinforcing steel. The design of the buildings was carried out by using the software ETABS (2016, Computers and Structures Inc., Berkeley, CA, USA) [50]. The buildings are assumed to be located in a zone in the valley of Mexico with a soil dominant period T

_{s}= 1.8 s (which falls in the intermediate zone as considered in the present study), with a seismic behavior coefficient, Q = 3 (which implies that the detailing requirements for their structural members are similar to those established by the Uniform Building Code [51] for reinforced concrete special moment-resisting frames). The design of the buildings was based on a modal spectral dynamic analysis. The lateral stiffness was selected so that the maximum inter-story drift does not exceed 0.030, which is the limit specified by MCBC-2004 for the design of ductile moment-resisting concrete frames.

_{1}), the resistant base shear (V

_{b}), the resistant base shear coefficient (c), the yield displacement (d

_{y}), and the ultimate displacement (d

_{u}), the latter two were obtained through nonlinear static (pushover) analyses.

#### 6.2. Ductility Transformation Factors (DTF)

## 7. Conclusions

- Results indicate that the additional lateral strength requirement of structures with asymmetric yielding is higher for those with fundamental vibration periods close to the dominant period of the soil where they are located; this requirement is even higher for structures located on soft soils.
- Simplified mathematical expressions were proposed for the estimation of strength amplification factors for structures with asymmetric yielding, considering different soil conditions. The expressions correspond to the valley of Mexico and depend on factors such as the ductility of the structure, the level of asymmetric yielding, and the ratio between the fundamental vibration period of the structure and that of the dominant period of the soil.
- The proposed mathematical expressions are more conservative than that recommended in the current Mexico City Building Code (MCBC-2004) for intermediate and soft soils, especially for structures whose vibration period is close to the dominant period of the soil where they are located. Although the expression proposed in the MCBC-2004 leads to conservative results for firm ground, results indicate that the effect of asymmetric yielding is much more detrimental on intermediate and soft soils than on firm ground. The expressions developed in this study have been approved by the Technical Committee for Seismic Design of the MCBC, and will be incorporated in the new version of the Mexico City Building Code.
- It was verified that the value of the expected ductility demand of asymmetric yielding MDOF systems, associated with a given return period, is almost equal to that corresponding to their equivalent simplified systems. The implication of this is that the use of simplified structural systems to estimate strength amplification factors for MDOF structures with asymmetric yielding is appropriate.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Some examples of structures with asymmetric yielding: (

**a**) asymmetry in vertical loads; (

**b**) structures with sloping facades; (

**c**) tilting.

**Figure 3.**Example of reinforced concrete tilted buildings located in the highlighted area of Figure 2.

**Figure 4.**(

**a**) plan and elevation, and (

**b**) definition of the parameter α of the simplified structural systems used in this study.

**Figure 6.**Pseudo-acceleration response spectra for 5% of critical damping, corresponding to the seismic records selected in (

**a**) Zone A, (

**b**) Zone B, (

**c**) Zone C, (

**d**) Zone D, (

**e**) Zone E, (

**f**) Zone F and (

**g**) Zone G of the valley of Mexico considered in this study (E-W Component).

**Figure 7.**Parameter R

_{BSCS}corresponding to (

**a**) Zone A; (

**b**) Zone B; (

**c**) Zone C; (

**d**) Zone D; (

**e**) Zone E; (

**f**) Zone F and (

**g**) Zone G, for different values of α, and µ = 2.

**Figure 8.**Comparison of the strength amplification factors (AF) obtained with Equations (6) and (7) with those presented in Figure 7, corresponding to a ductility demand µ = 2. (

**a**) Zone A; (

**b**) Zone B; (

**c**) Zone C; (

**d**) Zone D; (

**e**) Zone E; (

**f**) Zone F; (

**g**) Zone G.

**Figure 9.**Comparison of the strength amplification factors obtained with Equations (6) and (7) with those recommended by MCBC-2004, corresponding to µ = 2 and α = 0.02. (

**a**) Firm ground; (

**b**) Intermediate and soft soil.

**Figure 10.**Schematic representation related to the estimation of the ductility transformation factors (DTF).

**Figure 11.**Plan, elevation, and characterization of asymmetric yielding of the buildings. (

**a**) plan; (

**b**) elevation; (

**c**) asymmetric yielding building.

**Figure 12.**Ductility transformation factors between simplified systems (SS) and multi-degree of freedom (MDOF) systems for different values of α. (

**a**) Symmetric; (

**b**) α = 0.01; (

**c**) α = 0.02; (

**d**) α = 0.03.

Zone | Range of Period (s) | Average Dominant Period, T_{s} (s) |
---|---|---|

A | T_{s} ≤ 0.5 | 0.62 |

B | 0.5 < T_{s} ≤ 1.0 | 0.96 |

C | 1.0 < T_{s} ≤ 1.5 | 1.41 |

D | 1.5 < T_{s} ≤ 2.0 | 1.98 |

E | 2.0 < T_{s} ≤ 2.5 | 2.55 |

F | 2.5 < T_{s} ≤ 3.0 | 3.03 |

G | 3.0 < T_{s} ≤ 4.0 | 3.61 |

Zone | Dominant Period (s) | a | b | c | d |
---|---|---|---|---|---|

A | T_{s} ≤ 0.5 | (3.5µ − 1.5) α | 13.4 | 0.1 | 1.6α + 1 |

B | 0.5 < T_{s} ≤ 1.0 | (4.8µ − 3) α | 8.8 | 0.1 | 4.1α + 1 |

C | 1.0 < T_{s} ≤ 1.5 | (1.5µ − 1.4) α | 0.7 | 0.08 | 1 |

D | 1.5 < T_{s} ≤ 2.0 | (2µ − 1.6) α | 0.5 | 0.1 | 1 |

E | 2.0 < T_{s} ≤ 2.5 | (1.5µ + 0.8) α | 0.9 | 0.12 | 1 |

F | 2.5 < T_{s} ≤ 3.0 | (1.5µ + 1.1) α | 0.7 | 0.13 | 1 |

G | 3.0 < T_{s} ≤ 4.0 | (1.9µ − 0.05) α | 0.1 | 0.12 | 1 |

Building | T_{1} | V_{b} | W | c | d_{y} | d_{u} |
---|---|---|---|---|---|---|

(s) | (Ton) | (Ton) | (m) | (m) | ||

8-story | 1.2 | 1332.45 | 3807 | 0.35 | 0.168 | 0.63 |

9-story | 1.22 | 1498.2 | 4540 | 0.33 | 0.181 | 0.65 |

10-story | 1.33 | 1563.02 | 5042 | 0.31 | 0.195 | 0.69 |

11-story | 1.4 | 1721.1 | 5737 | 0.3 | 0.223 | 0.74 |

12-story | 1.48 | 1772.68 | 6331 | 0.28 | 0.246 | 0.78 |

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## Share and Cite

**MDPI and ACS Style**

Valenzuela-Beltrán, F.; Ruiz, S.E.; Reyes-Salazar, A.; Gaxiola-Camacho, J.R. On the Seismic Design of Structures with Tilting Located within a Seismic Region. *Appl. Sci.* **2017**, *7*, 1146.
https://doi.org/10.3390/app7111146

**AMA Style**

Valenzuela-Beltrán F, Ruiz SE, Reyes-Salazar A, Gaxiola-Camacho JR. On the Seismic Design of Structures with Tilting Located within a Seismic Region. *Applied Sciences*. 2017; 7(11):1146.
https://doi.org/10.3390/app7111146

**Chicago/Turabian Style**

Valenzuela-Beltrán, Federico, Sonia E. Ruiz, Alfredo Reyes-Salazar, and J. Ramón Gaxiola-Camacho. 2017. "On the Seismic Design of Structures with Tilting Located within a Seismic Region" *Applied Sciences* 7, no. 11: 1146.
https://doi.org/10.3390/app7111146