# Probabilistic Seismic Assessment of Existing Masonry Buildings

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Assessment of the Seismic Performance of Existing Masonry Buildings

#### 2.1. The E-PUSH Program

- only lateral stiffness of masonry walls is taken into account, disregarding transverse stiffness;
- horizontal force-lateral displacement diagram of the wall is elastic-plastic, with the plastic plateau limited by the elastic drift ${\delta}_{e}$ and the ultimate drift ${\delta}_{u}$.
- verification is performed in terms of seismic capacity and demand using the acceleration-displacement response spectra (ADRS) taking into account the story displacements.

- $\delta <{\delta}_{e}$, the wall is still in the elastic range;
- ${\delta}_{e}\le \delta \le {\delta}_{u}$, the wall is in the plastic range, it achieved its shear force resistance and its equivalent stiffness is reduced;
- $\delta >{\delta}_{u}$, the wall sustains only vertical loads and its shear resistance and its lateral stiffness are set to zero.

- the non-linear capacity curve of the structure is transformed in an equivalent bi-linear elastic-plastic curve as described in [14]; the curve is characterized by the maximum force ${F}_{y,eq}$, calculated averaging the maximum base shear in the non-linear capacity curve and the base shear corresponding to the attainment of the elastic limit; by the yield displacement ${\delta}_{y,eq}$, evaluated as the ratio between ${F}_{y,eq}$ and the effective stiffness of the structure ${K}_{E}$; and by the ultimate displacement ${\delta}_{u,eq}$ which is the maximum displacement in the capacity curve;
- the bi-linear, force-displacement capacity curve, $F-\delta $, of the multi-degree of freedom (MDOF) system is converted in an acceleration-displacement, ${S}_{a}-{S}_{d}$, capacity diagram, for an equivalent single degree of freedom (SDOF) system, according to the following formulae$${S}_{a}=\frac{F}{\mathsf{\Gamma}{m}^{*}};\text{}{S}_{d}=\frac{\delta}{\mathsf{\Gamma}};$$$$\mathsf{\Gamma}=\frac{{{\displaystyle \sum}}_{j=1}^{N}{m}_{j}{\varphi}_{j}}{{{\displaystyle \sum}}_{j=1}^{N}{m}_{j}{{\varphi}_{j}}^{2}}$$$${m}^{*}={\displaystyle \sum}_{j=1}^{N}{m}_{j}{\varphi}_{j},$$

- 3.
- in order to obtain the demand diagram, the elastic design spectrum defined in the Italian Building code [15] from the standard pseudo acceleration-natural period, ${S}_{ae}-T$ is converted into the pseudo acceleration-displacement format ${S}_{ae}-{S}_{de}$ through:$${S}_{de}=\frac{{T}^{2}}{4{\pi}^{2}}{S}_{ae};$$
- 4.
- the capacity spectrum and demand spectrum curves are plotted in the same graph, to define displacement demand. If the capacity curve intersects the demand curve, the displacement demand is assumed equal to the intersection point ${d}_{t}$. Otherwise, the displacement demand is determined starting from the intersection of the radial line corresponding to the elastic period ${T}^{*}$ of the structure with the elastic design spectrum defining the acceleration demand ${S}_{ae}\left({T}^{*}\right)$. A reduction factor ${R}_{\mu}$ is defined as the ratio between the acceleration demand ${S}_{ae}\left({T}^{*}\right)$ and the yield acceleration S
_{a,y.}$${R}_{\mu}=\frac{{S}_{ae}\left({T}^{*}\right)}{{S}_{a,y}}.$$

- 5.
- the seismic performance of the structure is evaluated comparing the displacement demand ${d}_{t}$ with the ultimate displacement defined by the capacity curve ${d}_{c}$.

- the bi-linear capacity curve of the structure (red solid line);
- the design spectrum $SL{V}_{e}$ (blue solid curve) for a return period of 712 years (${T}_{RD}$), referring to the ultimate limit state for life safety ($SLV$) of occupants in a given location (Florence Municipality in this case);
- the design inelastic spectrum $SL{V}_{a}$ (blue dashed curve);
- the displacement demand ${d}_{t}$ (scarlet dashed line) to be compared with the ultimate displacement ${d}_{c}$ (green dashed line);
- the elastic spectrum (green solid curve) for the return period T
_{RC}consistent with the capacity of the structure (${d}_{t}={d}_{c}$) and the corresponding inelastic spectrum (green dashed curve).

#### 2.2. Validation of the E-PUSH Program

#### 2.3. Challenges in the Assessment of the Seismic Performance

## 3. Case Studies

^{3}and 45,000 m

^{3}) and inter-story heights (height varies between 3 m and 5 m).

## 4. Methodology

#### 4.1. Identification of Uncertain Parameters

- the shear modulus G and the elastic modulus E of masonry;
- the shear strength ${\tau}_{k}\text{}$;
- the ultimate displacement ${\delta}_{u}$.

#### 4.2. Sensitivity Analysis

#### 4.2.1. Response Surface via generalized Polynomial Chaos Expansion

**u**is a vector that gathers the response quantities, in this case the seismic risk index I

_{R}

_{.}

**u**replacing the $\tilde{G}$ map can be of great use. Taking advantage of functional approximations of the random variables by means of the generalized polynomial chaos (gPC) expansion, the construction of response surfaces is significantly facilitated and a computationally cheap proxy model is obtained [38]. For an extensive review of this topic, please refer to [39].

**u**in terms of

**z**, with the advantage that for any realization of the random vector $\mathit{Q}$ the response u can be easily evaluated by first mapping

**q**to

**z**and then evaluating the gPC expansion without much computational expense.

- in terms of $G/{\tau}_{k}$ and ${\delta}_{u}$, assuming mean values for ${\tau}_{k}$ and $G/E$;
- in terms of $G/{\tau}_{k}$ and ${\tau}_{k}$, assuming mean values for ${\delta}_{u}$ and $G/E$;
- in terms of ${\tau}_{k}$ and ${\delta}_{u}$, assuming mean values for $G/{\tau}_{k}$ and $G/E$;
- in terms of $G/{\tau}_{k}$ and $G/E$ assuming mean values for ${\tau}_{k}$ and ${\delta}_{u}$.

#### 4.2.2. Evaluation of Sobol Indices

#### 4.3. Uncertainty Quantification

## 5. Seismic Performance Classification

_{R}

_{,50}) and the 5th percentile (I

_{R}

_{,05}) of the seismic risk index. In particular, five categories (A–E) with increasing magnitude of vulnerability are defined according the interval reported in Table 3.

## 6. Conclusions

_{R}through gPC-expansion; in this way, quantification of uncertainties in the resulting seismic risk index is easily obtained, reducing the computation effort, and Sobol sensitivity indices can be computed analytically from the gPCE coefficients.

_{R}

_{,50}) and the 5th percentile (I

_{R}

_{,05}) of the seismic risk index.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Final output of the pushover analysis, verification in the acceleration-displacement response spectra (ADRS) plane.

**Figure 2.**Verification in the ADRS plane parameterized as a function of the damping coefficient $\xi $.

**Figure 4.**Sketch of the two story masonry structure tested at Georgia Tech laboratory [22], plan view and elevation.

**Figure 6.**Comparison of capacity curves of walls A and B with experimental $H-\delta $ test results [22].

**Figure 7.**Comparison of capacity curves of walls 1 and 2 with experimental $H-\delta $ test results [22].

**Figure 8.**Three-dimensional (3D) layout of the resistant shear walls for the investigated masonry buildings.

Random Variable | $\overline{\mathit{X}}$ | COV |
---|---|---|

${\tau}_{k}$ | Mean Value for each masonry class in [29], e.g., 0.026 N/mm^{2} for stone | 0.14 |

$G/{\tau}_{k}$ | 1500 | 0.3 |

$G/E$ | 0.15 | 0.2 |

${\delta}_{u}$ | 0.004 | 0.2 |

School Building | ${\mathit{I}}_{\mathit{R},05}$ | ${\mathit{I}}_{\mathit{R},50}$ | ${\mathit{I}}_{\mathit{R},95}$ |
---|---|---|---|

B1 | 0.40 | 0.52 | 0.62 |

B2 | 0.37 | 0.54 | 0.73 |

B3 | 0.81 | 1.37 | 1.48 |

B4 | 0.37 | 0.53 | 0.74 |

B5 | 0.34 | 0.48 | 0.67 |

B6 | 0.99 | 1.45 | 1.48 |

B7 | 0.43 | 0.53 | 0.62 |

B8 | 0.79 | 1.37 | 1.48 |

B9 | 0.51 | 0.75 | 0.94 |

B10 | 0.39 | 0.52 | 0.67 |

B11 | 0.48 | 0.74 | 1.01 |

${\mathit{I}}_{\mathit{R},05}\text{}$ | ||||||
---|---|---|---|---|---|---|

<0.2 | 0.2–0.3 | 0.3–0.5 | 0.5–0.7 | >0.7 | ||

${I}_{R,50}$ | <0.3 | E | E | - | - | - |

0.3–0.6 | E | D | D | - | - | |

0.6–0.8 | E | D | C | C | - | |

0.8–1 | D | C | C | B | B | |

>1 | D | C | B | B | A |

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**MDPI and ACS Style**

Croce, P.; Landi, F.; Formichi, P.
Probabilistic Seismic Assessment of Existing Masonry Buildings. *Buildings* **2019**, *9*, 237.
https://doi.org/10.3390/buildings9120237

**AMA Style**

Croce P, Landi F, Formichi P.
Probabilistic Seismic Assessment of Existing Masonry Buildings. *Buildings*. 2019; 9(12):237.
https://doi.org/10.3390/buildings9120237

**Chicago/Turabian Style**

Croce, Pietro, Filippo Landi, and Paolo Formichi.
2019. "Probabilistic Seismic Assessment of Existing Masonry Buildings" *Buildings* 9, no. 12: 237.
https://doi.org/10.3390/buildings9120237