# Nonlinear Seismic Analysis of Existing RC School Buildings: The “P3” School Typology

^{*}

## Abstract

**:**

## 1. Introduction

_{I}) for this type of buildings, which multiplies the reference acceleration (a

_{gR}). An increase of the importance factor causes an increase of the return period (T

_{R}) of the seismic action, and, consequently, an increase of the vibration level that the buildings should be able to resist. Unfortunately, many school buildings are still collapsing or being severely damaged after the occurrence of worldwide destructive earthquakes [5,6,7,8,9], including reinforced concrete (RC) constructions; such was the case of the School “Enrique Rebsamen” in Mexico [10].

## 2. The “P3” School Typology

## 3. Nonlinear Static Analysis

#### 3.1. The Characteristics of the Structural Elements

#### 3.2. Simulated Design of RC Beams

#### 3.3. The Obtained Capacity Curves

_{b}) and the displacements of the control node (d

_{n}), normally being the mass centre (MC) of the roof top.

_{i}associated to each degree of freedom i) in an equivalent single degree of freedom problem (SDOF), with stiffness k* and mass m*, by using a transformation coefficient (Γ).

_{i}), which then is normalized so that the value ϕ

_{n}of the control node is unitary, being

_{0i}) was applied to the N degrees of freedom of each structural school module. These forces were determined so that the sum of those forces was equal to the unity, being

_{b}was equal to the load parameter (λ) computed by the SeismoStruct during the nonlinear structural analysis.

_{i}= 1, and a “modal” distribution, which in the present study was considered equal to shape deformation associated with the vibration mode with the highest participation factor in each direction), as schematized in Figure 9.

## 4. Seismic Safety Evaluation

_{gR}that is established for the municipalities of the Algarve region, the increase of the soil factor (S), and because the period T

_{C}presents high values, increasing the plateau of the maximum spectral acceleration despite the design seismic action of the RSAEEP being multiplied by a factor of 1.5 (as defined in the code), which can be observed in Figure 13 for the two earthquake types that are established in both seismic codes.

_{LR}), which is 475 years in the NP EN 1998-1:2010. However, the damage evaluation of the existing school buildings was carried out considering the philosophy that is presented in the EC8-3, where the seismic safety is evaluated based on several limit states (LS), which are associated with different return periods for the seismic action. Moreover, EC8-1 indicates a simplified expression for the determination of a factor (γ

_{L}), which can be used to quantify the reference acceleration associated with any other return period (a

_{gRL}):

_{gRL}) by using Equation (5) for all the LS that are established in the EC8-3.

#### 4.1. Limit States

#### 4.2. Performance Points and Performance Curves

_{R}= 475 years), which are associated to the performance points, even exceeded the values corresponding to the NC limit state.

_{gR}by a coefficient (γ

_{LS}), between 0 and 1 (Figure 16), which is computed using the expression (6).

_{a}* in a fast way, for all the displacements until collapse, without any iterative procedure. For that purpose, the period T* is first compared with the period T

_{C}, that is established in the EC8-1 for each type of seismic action and ground type, in order to define which mathematical expression must be used to compute S

_{a}*.

_{C}, then

_{C}, then

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Primary school buildings existing in the regions of Algarve (Portugal) and Huelva (Spain).

**Figure 2.**Scheme of the classroom modules of the “P3” schools: (

**a**) two classrooms placed in one floor; (

**b**) three classrooms placed in one floor; (

**c**) six classrooms placed in two floors; (

**d**) four classrooms placed in two floors.

**Figure 3.**Example of different module combinations presented in some “P3” schools that are still functioning as primary schools in the Algarve region in: (

**a**) Silves; (

**b**) Marmelete, Monchique; (

**c**) Faro; and (

**d**) Portimão.

**Figure 4.**Architectural plans of the primary school EB1 n.5 of Faro, with the location of the reinforced concrete (RC) columns: (

**a**) ground floor; (

**b**) first floor.

**Figure 7.**SAP2000 structural models adopted for the simulated design of the RC beams and stairs: (

**a**) modules 1 and 2; (

**b**) module 3; (

**c**) module 4.

**Figure 8.**SeismoStruct structural models adopted for obtaining the capacity curves: (

**a**) modules 1 and 2; (

**b**) module 3; (

**c**) module 4.

**Figure 10.**Capacity curves determined for the school modules 1 and 2 (M—modal; U—uniform force pattern): (

**a**) X direction; (

**b**) Y direction.

**Figure 11.**Capacity curves determined for the school module 3 (M—modal; U—uniform force pattern): (

**a**) X direction; (

**b**) Y direction.

**Figure 12.**Capacity curves determined for the school module 4 (M—modal; U—uniform force pattern): (

**a**) X direction; (

**b**) Y direction.

**Figure 13.**Response spectra for the seismic actions that are now established in the Portuguese National Annex of the EC8-1, for school buildings placed in Faro on a ground type C, and the corresponding seismic actions that were established in the national seismic code (RSAEEP) that was mandatory when the studied “P3” school was built: (

**a**) far-field offshore earthquake scenario (type 1); (

**b**) near-source earthquake scenario (type 2).

**Figure 15.**Example of the output of the developed software that computes the performance points for: (

**a**) earthquake type 1; (

**b**) earthquake type 2.

**Figure 16.**Flowchart of the adopted process for obtaining a performance curve for a given capacity curve and response spectrum.

**Figure 17.**Worst-case performance curves determined for each direction of the structural independent modules of the studied “P3” school, and the seismic action levels that are established in the Portuguese National Annex [35] of the EC8-3 for DL, SD and NC limit states.

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**MDPI and ACS Style**

Estêvão, J.M.C.; Esteves, C.
Nonlinear Seismic Analysis of Existing RC School Buildings: The “P3” School Typology. *Buildings* **2020**, *10*, 210.
https://doi.org/10.3390/buildings10110210

**AMA Style**

Estêvão JMC, Esteves C.
Nonlinear Seismic Analysis of Existing RC School Buildings: The “P3” School Typology. *Buildings*. 2020; 10(11):210.
https://doi.org/10.3390/buildings10110210

**Chicago/Turabian Style**

Estêvão, João M.C., and Carlos Esteves.
2020. "Nonlinear Seismic Analysis of Existing RC School Buildings: The “P3” School Typology" *Buildings* 10, no. 11: 210.
https://doi.org/10.3390/buildings10110210