On the Seismic Design of Structures with Tilting Located within a Seismic Region
Buildings with Asymmetric Yielding Produced by Tilting in Mexico City
2. Simplified Structural Models
3. General Methodology
3.1. Evaluation of the Structural Reliability
3.2. Steps to Follow for the Proposed Methodology
- 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 (T1). 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 (µx, µy) considering the simultaneous action of both horizontal components of the seismic ground motions; µx and µy are calculated by taking the maximum horizontal displacement of the center of mass of the structural systems in each direction (dX, dY) divided by their yield displacement, dy, 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).
- Next, the median (D) and standard deviation (σlnD) of the ductility demand logarithms are calculated.
- Fragility curves for several values of the maximum ductility demand are obtained using Equation (3):
- Ductility demand hazard curves (DDHC, Equation (1)) are obtained for symmetric yielding systems, and alternatively, for systems with different levels of asymmetric yielding.
- 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.
- 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 T1 and c, corresponding to a given µ-UERS. The process is repeated several times in order to obtain a data set T1 vs c for each value of the ductility demand considered. More details about this process can be found in . Ratios between BSCS of systems with different levels of asymmetric yielding with respect to symmetric systems, RBSCS, 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:
- 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.
- 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
5. Mathematical Expressions of Strength Amplification Factors Corresponding to the Seismic Region of Interest
Comparison of the Proposed Mathematical Expressions with that Recommended in the Current Mexico City Building Code
6. Ductility Transformation Factors between Simplified and MDOF Systems
6.1. Characteristics of the Buildings Analyzed
6.2. Ductility Transformation Factors (DTF)
- 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.
Conflicts of Interest
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|Zone||Range of Period (s)||Average Dominant Period, Ts (s)|
|A||Ts ≤ 0.5||0.62|
|B||0.5 < Ts ≤ 1.0||0.96|
|C||1.0 < Ts ≤ 1.5||1.41|
|D||1.5 < Ts ≤ 2.0||1.98|
|E||2.0 < Ts ≤ 2.5||2.55|
|F||2.5 < Ts ≤ 3.0||3.03|
|G||3.0 < Ts ≤ 4.0||3.61|
|Zone||Dominant Period (s)||a||b||c||d|
|A||Ts ≤ 0.5||(3.5µ − 1.5) α||13.4||0.1||1.6α + 1|
|B||0.5 < Ts ≤ 1.0||(4.8µ − 3) α||8.8||0.1||4.1α + 1|
|C||1.0 < Ts ≤ 1.5||(1.5µ − 1.4) α||0.7||0.08||1|
|D||1.5 < Ts ≤ 2.0||(2µ − 1.6) α||0.5||0.1||1|
|E||2.0 < Ts ≤ 2.5||(1.5µ + 0.8) α||0.9||0.12||1|
|F||2.5 < Ts ≤ 3.0||(1.5µ + 1.1) α||0.7||0.13||1|
|G||3.0 < Ts ≤ 4.0||(1.9µ − 0.05) α||0.1||0.12||1|
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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
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/app7111146Chicago/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