# Uncertainty in Seismic Capacity of Masonry Buildings

^{*}

## Abstract

**:**

## 1. Introduction

_{p}associated with peak strength σ

_{p}is less uncertain than ultimate strain ε

_{u}and residual strength σ

_{u}. Strain softening of masonry induces lower confidence in the simulation of plastic behavior given that the fracture process is noticeably sensitive to local material discontinuities. For instance, pumice inclusions and voids can be detected in tuff stones, which are widely spread in many earthquake-prone regions. Compressive tests on tuff masonry in the direction orthogonal to bed joints [17] have shown a coefficient of variation (CoV) equal to 12.5% and 9.1% for σ

_{p}and ε

_{p}, respectively; such values increase even to 23.4% and 73.1% for strength and limit strain at collapse. Young’s and shear moduli at σ

_{p}/3 have been found to have CoV equal to 16.7% and 17.7%, respectively, whereas CoV = 26.8% has been computed for the relevant Poisson’s ratio. Different CoV-values have been detected under compression parallel to bed joints. The representation of confidence interval under varying axial strain has confirmed that uncertainty increases with inelasticity. It is emphasized that such an outcome depends on the randomness of fracture process and the ability of the test setup in capturing cracks. Figure 1 also shows probability density functions (PDFs) of peak and ultimate strengths, as well as conditional PDFs of cracking and ultimate strains. This is a typical representation for strains’ uncertainties because they are usually estimated at given strength levels. Ultimate strain is associated with a predefined strength drop.

## 2. Accounting for Uncertainty in Seismic Capacity Estimation of a Case-Study Masonry Building

_{1}= 0.24 s through a simplified code formula [21]. A simulated design procedure was employed in accordance to Eurocode 8 (EC8)—Part 3 [22] on the basis of practice rules, without considering specific seismic design provisions.

^{2}in plan and 8.16 m high, given that interstory height was set to 4.08 m.

#### 2.1. Material Properties and Related Uncertainty

_{m}); shear strength at zero confining stress (τ

_{0}); Young’s modulus (E

_{m}); shear modulus (G

_{m}); and available ductility in shear (µ

_{s}). Lower and upper bounds, mean and standard deviation of such modeling variables are listed in Table 1, where one can note that CoV was equal to: 15% in the case of f

_{m}, E

_{m}, and G

_{m}; 12.19% in the case of τ

_{0}; and 23.90% in the case of µ

_{s}. Young’s modulus was about 1000 times compressive strength, whereas shear modulus was 0.4 times Young’s modulus. According to EC6 [23], friction coefficient of masonry was assumed to be 0.4 at any confining stress level. The spatial distribution of material properties was assumed to be uniform throughout the building and wall sizes were assumed to be deterministically known. Realizations of the random modeling vector Θ = [γ, f

_{m}, τ

_{0}, E

_{m}, G

_{m}, µ

_{s}] were generated through the full Monte Carlo method.

Parameter | γ [kN/m^{3}] | f_{m} [MPa] | τ_{0} [MPa] | E_{m} [MPa] | G_{m} [MPa] | µ_{s} |
---|---|---|---|---|---|---|

Lower bound | 14.00 | 1.40 | 0.038 | 1746 | 698 | 2.64 |

Upper bound | 18.00 | 2.40 | 0.057 | 2444 | 978 | 3.71 |

Mean | – | 1.90 | 0.047 | 2090 | 836 | 3.18 |

Standard deviation | – | 0.28 | 0.006 | 314 | 125 | 0.76 |

#### 2.2. Capacity Modeling

_{s}was used to amplify the total displacement of the macroelement in the case of shear failure.

## 3. Static Pushover Analysis

_{a}versus roof lateral displacement d

_{c}up to the collapse of the building, which was assumed to occur when all piers minus one at the same story reached their displacement capacity. It is emphasized that displacement capacity of the whole building is different from that of a macroelement, opposed to the case of other structural systems (e.g., RC frames) where the attainment of the limit chord rotation in a member is also assumed to be a global limit state. In the case of masonry constructions, the building displacement capacity at life safety (d

_{u}) is typically defined as the lateral displacement of a control point on the roof corresponding to 20% resistance drop on the post-peak descending branch of the SPO curve. Therefore, one can estimate spectral acceleration capacity S

_{a}|d

_{u}as the intensity measure causing the life safety limit state.

**Figure 5.**Static pushover (SPO) curves and their means related to the analysis cases: (

**a**) 1X; (

**b**) 1Y.

_{e}is the limit elastic displacement, S

_{ae}is the spectral acceleration associated with d

_{e}, and Γ

_{1}is the first-mode participation factor of the structure. Therefore, T

_{e}is about one-fourth of the fundamental period T

_{1}estimated in accordance with EC8 [21] (see Section 2). This produces a low deformation capacity, but also a limited displacement demand on the structure. Note that the mean limit elastic displacement of the building is 0.6 and 0.2 mm in the X- and Y-direction, respectively.

## 4. Estimation of Seismic Capacity Uncertainty

_{c}, denoted by µ

_{Sa}

_{|dc}and σ

_{Sa}

_{|dc}respectively, were investigated for each loading direction (Figure 7 and Figure 8). It is confirmed that the earthquake resistance of the case-study building is not the same for both loading directions and lateral load patterns. In addition, different SPO curves were obtained for positive and negative orientations of seismic actions, especially in the case of the X-direction. This was caused by the lack of symmetry in plan which induced higher seismic response sensitivity to axial forces in piers generated by overturning moments at each floor level. Variations in axial forces produce a change in lateral capacity of pier panels, as analytically shown by Augenti [10]. It is worth noting that dispersion of conditional spectral acceleration increases until collapse of some macroelements is attained, Figure 8.

_{Sa}

_{|dc}reaches a constant value falling in the ranges [0.04 g, 0.09 g] and [0.02 g, 0.03 g] for seismic actions acting in the X- and Y-direction, respectively. This also indicates a reliable estimation of seismic capacity uncertainty, which is quantified in Table 2. In particular, one can note that CoV is 9%–26% in the X-direction and 12%–55% in the Y-direction, in the case of d

_{u}. Uncertainty reduces in the case of spectral acceleration capacity S

_{a}|d

_{u}for which CoV is 8%–11% in the X-direction and 11%–24% in the Y-direction. For each code-based load combination, uncertainty in the Y-direction is higher than that in the other direction of the building plan.

Analysis Case | Load Pattern | μ_{d} [mm] | CoV_{d} | µ_{Sa|du} [g] | CoV_{Sa|du} |
---|---|---|---|---|---|

1X | ULP | 0.77 | 12% | 0.61 | 8% |

ITLP | 0.76 | 11% | 0.42 | 11% | |

2X | ULP | 0.77 | 11% | 0.61 | 8% |

ITLP | 0.75 | 9% | 0.41 | 9% | |

3X | ULP | 1.51 | 22% | 1.02 | 9% |

ITLP | 1.28 | 19% | 0.57 | 10% | |

4X | ULP | 1.30 | 26% | 1.06 | 8% |

ITLP | 1.32 | 21% | 0.55 | 9% | |

1Y | ULP | 0.29 | 12% | 0.20 | 11% |

ITLP | 0.21 | 25% | 0.11 | 16% | |

2Y | ULP | 0.28 | 13% | 0.20 | 11% |

ITLP | 0.24 | 29% | 0.11 | 18% | |

3Y | ULP | 0.20 | 55% | 0.16 | 19% |

ITLP | 0.18 | 51% | 0.09 | 24% | |

4Y | ULP | 0.26 | 51% | 0.20 | 11% |

ITLP | 0.18 | 37% | 0.11 | 20% |

_{θ}(θ,Θ

_{f}) where the vector Θ

_{f}included the mean µ

_{θ}and standard deviation σ

_{θ}estimated over experimental data available in the literature. Actually, such properties depend on mechanical parameters and relative size of masonry constituents (i.e., masonry units and mortar joints), as well as masonry bond and boundary conditions [10,11]. This means that, in line of principle, the overall uncertainty could be reduced assuming that it is partly aleatory and partly epistemic [26]. Aleatory uncertainty cannot be reduced by increasing the size of the data set of modeling parameters or modeling rules, because it is assumed to be dependent on the randomness of the phenomenon to be modeled. Conversely, epistemic uncertainty is treated as a variable which decreases as the knowledge level increases [27]. In the case of existing building, material and geometric properties are epistemic variables, whereas loads and their associated demands on the structure, be it existing or future, have always future realizations so their uncertainty is typically classified as aleatory. An important source of epistemic uncertainty is also due to the use of a physical structural model which is a simplification of the real construction. On the material side, experimental data sets on parameters of both masonry assemblages and their constituents are still lacking to be reliably used, especially in the case of tuff masonry. This causes the following missing submodels: (1) physical submodels θ = g(β,Θ

_{g}) + ε describing each tuff masonry property θ as a function of the vector β of geometric and mechanical parameters of masonry constituents, which is characterized by the vector of parameters Θ

_{g}fitted to experimental data and statistical uncertainty (residual) ε; and (2) probabilistic submodels of masonry constituent properties f

_{β}(β,Θ

_{f}). When such an information will be available, the use of Monte Carlo-type or other methods to obtain realizations of masonry constituent properties will enable to reduce uncertainty in tuff masonry properties, and hence in seismic capacity of tuff masonry structures. Nevertheless, at this state of knowledge, the identification of epistemic variables ensures transparency in the quantitative analysis presented above, because one can know which reducible uncertainties have been left unreduced in the modeling phase. The present paper thus provides a first estimation of seismic capacity of a typical existing URM building which could be improved in the future on the basis of more observed data on mechanical properties.

## 5. Conclusions

_{C}or elastic spectral acceleration S

_{ae,C}), as well as the corresponding probability of failure P

_{f}in a given time interval, which are crucial for decision-making in PBEE. This computation should include the uncertainty in seismic demand, which has not been estimated in this study. To this end, in the context of SPO-based approaches, two alternative procedures are available to determine the relationship between IM and a given engineering demand parameter (EDP), e.g., the roof displacement d

_{c}, without using incremental dynamic analysis (IDA) [28] for a large suite of earthquake ground motion records: SPO2IDA tool [29,30] and incremental N2 (IN2) method [31]. Both methods could provide approximate IDA curves, but different procedures should be employed to estimate central values (i.e., mean or median) and dispersion in seismic performance, accounting for aleatory uncertainty due to record-to-record variability. Then, the inclusion of seismic hazard uncertainties could allow to fully estimate P

_{f}under varying IM, deriving the fragility curve at collapse. Nevertheless, further research is needed to assess the applicability of SPO2IDA tool and IN2 method to URM buildings. Furthermore, extensive sensitivity studies will have to be performed in order to define reliable default values for dispersion measures associated with displacement demand.

## Acknowledgments

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**MDPI and ACS Style**

Parisi, F.; Augenti, N.
Uncertainty in Seismic Capacity of Masonry Buildings. *Buildings* **2012**, *2*, 218-230.
https://doi.org/10.3390/buildings2030218

**AMA Style**

Parisi F, Augenti N.
Uncertainty in Seismic Capacity of Masonry Buildings. *Buildings*. 2012; 2(3):218-230.
https://doi.org/10.3390/buildings2030218

**Chicago/Turabian Style**

Parisi, Fulvio, and Nicola Augenti.
2012. "Uncertainty in Seismic Capacity of Masonry Buildings" *Buildings* 2, no. 3: 218-230.
https://doi.org/10.3390/buildings2030218