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Journal Article•DOI•

Fragility Functions for Tall URM Buildings around Early 20th Century in Lisbon, Part 2: Application to Different Classes of Buildings

Ana Simões1, Rita Bento1, Sergio Lagomarsino2, Serena Cattari2, Paulo B. Lourenço3 •
Instituto Superior Técnico1, University of Genoa2, University of Minho3
04 Mar 2021-International Journal of Architectural Heritage (Informa UK Limited)-Vol. 15, Iss: 3, pp 373-389
TL;DR: In this paper, the application of the procedure for the derivation of fragility functions presented in the companion article entitled Fragility functions for tall URM buildings around early 1990s is described.
Abstract: This article describes the application of the procedure for the derivation of fragility functions presented in the companion article entitled Fragility functions for tall URM buildings around early...

Summary (4 min read)

Jump to: [1. Introduction] – [2.1. Identification of classes of buildings] – [2.2. Logic-Tree Approach: epistemic uncertainties] – [3.1. Sensitivity analyses to reduce the number of classes of buildings] – [3.2. Monte Carlo Simulations: aleatory uncertainties] – [3.3. Non-linear dynamic analyses to validate the load pattern adopted on the non-linear static (pushover) analyses] – [4.1. Class of buildings H-S-S-S-H] – [4.1.1. Global behaviour] – [4.1.2. Local behaviour] – [4.2. Comparison between classes of buildings] – [5. Fragility curves for the typical URM tall buildings] and [6. Final Remarks]

1. Introduction

  • Seismic vulnerability addresses the susceptibility to suffer damage or loss due to an earthquake.
  • Basically, it comprehends the generation of fragility functions for the in-plane and out-of-plane response following different criteria and methods of analyses.
  • The intensity measure that produces the attainment of the limit state is obtained from the application of the Capacity-Spectrum Method with overdamped spectrum, as proposed in (Lagomarsino and Cattari, 2015).
  • These features are considered as epistemic uncertainties (related to the incomplete knowledge) and are treated by the logic-tree approach.
  • Finally, the fragility curves determined for the different classes of buildings, selected as the “minima” required to obtain a reliable description of the behaviour of the structural typology under examination, are compared and the final fragility curves for these unreinforced masonry buildings are provided.

2.1. Identification of classes of buildings

  • The development of vulnerability models at territorial scale firstly requires the identification of classes of buildings.
  • Thus, the features that mostly affect the seismic performance must be singled out depending on the characteristics of the building stock.
  • The ground floor level may be used as housing or as shop, implying a different layout of openings as exemplified in Figure 1.
  • During this period of construction in Lisbon, the URM buildings may include rubble stone masonry walls, clay brick (solid or hollow units) masonry walls and timber walls.
  • Therefore, it may be assumed that the side walls of 70% of the buildings are made of solid clay brick masonry, while the remaining 30% are made of solid clay brick masonry in the first three floors and hollow clay brick masonry in the last two floors.

2.2. Logic-Tree Approach: epistemic uncertainties

  • From the data illustrated in §2.1, the main variations identified for the tall URM “gaioleiro” buildings are summarized as: 1. Ground floor configuration: use of the building as i) housing (H) or as ii) shop (S).
  • I) solid clay brick masonry (S) or ii) solid clay brick masonry in the first two floors and hollow clay brick masonry in the last three floors (SH), also known as Side walls material.
  • These variations are considered as epistemic uncertainties and are organized in a logic-tree.
  • Each node of the tree represents a specific feature of the buildings and each branch represents an alternative option associated to that feature.
  • The logic-tree proposed for the tall URM “gaioleiro” buildings is presented in Figure 3, comprising a total of 32 building classes (identified as group A).

3.1. Sensitivity analyses to reduce the number of classes of buildings

  • Non-linear static analyses are carried out to the 32 classes of buildings models defined by the epistemic uncertainties (Figure 3) and by the median properties of the aleatory variables identified in the companion article (Simões et al., 2019a).
  • The variations are higher in the X direction than in the Y direction, but in all cases the Coefficient of Variation (CoV) is lower than 10%.
  • The final 8 classes of buildings are presented in the logic-tree from Figure 3 and identified as group B. Figure 8 presents the pushover curves for the 8 final classes of buildings obtained with the application of the uniform load distribution.
  • The seismic behaviour in the X direction is, in general, characterized by: i) lower stiffness and strength for classes of buildings with timber walls (#-#-SH-T-T) in comparison with the ones without such walls (#-#-S-S-H); ii) lower stiffness and strength for classes of buildings with independent side walls (#-I-#-#-#) in comparison with shared side walls (#-S-#-#-#).

3.2. Monte Carlo Simulations: aleatory uncertainties

  • The dispersion of the fragility curve takes into account also the aleatory variability in the definition of the capacity (C).
  • The latter is related to the random/aleatory variables aiming to account for both the uncertainties in the quantification of specific parameters and the intrinsic variations between buildings.
  • A total of 50 aleatory variables are considered for the analysis of the global behaviour (Simões et al., 2019a).
  • The Monte Carlo Method (Rubinstein, 2011) is used to define possible outcome values for each aleatory variables.
  • As the objective is not to obtain the tail of the distributions (i.e. to estimate rare events), the number of required simulations is small and the use of other methods such as Latin Hypercube or importance sampling are not required.

3.3. Non-linear dynamic analyses to validate the load pattern adopted on the non-linear static (pushover) analyses

  • The selection of the load distributions to perform non-linear static analyses is a critical issue.
  • Each of the 30 response spectra is defined by the acceleration spectra associated to the geometric mean of the two horizontal components affected by the scale factor.
  • Non-linear dynamic analyses are carried out to the 8 classes of buildings defined by the median properties of the aleatory variables, considering the records compatible with code seismic action type 1 and 2 for Lisbon applied in X and Y directions of the structure.
  • The effects of viscous damping are considered by adopting the Rayleigh damping formulation and assuming a viscous damping constant and equal to 3%.
  • Since the 30 records were scaled in the range of periods of the structure and not to the same value of peak ground acceleration (PGA), some records produce a quite linear response of the structure while others highlight the strong non-linear response until the collapse of the structure.

4.1. Class of buildings H-S-S-S-H

  • The procedure adopted for the derivation of fragility functions was illustrated in (Simões et al., 2019a) for the class of buildings H-S-S-S-H.
  • In the following sections, additional information is provided to support the determination of the fragility functions corresponding to the in-plane response (global behaviour, §4.1.1) and outof-plane response (local behaviour, §4.1.2).

4.1.1. Global behaviour

  • Non-linear static pushover analyses are carried out to the 39 building models defined by the aleatory variables.
  • As expected, these are not positioned in the middle of the cloud of results.
  • In the X direction, wall-2 (street façade) has a uniform deformation, while the remaining walls have a soft-storey mechanism.
  • The intensity measure compatible with PL, IMPL, is obtained from the application of the Capacity-Spectrum Method with overdamped spectrum and without any iterative procedure, as proposed in (Lagomarsino and Cattari, 2015).
  • These fragility curves are derived by considering the minimum between the fragility curves obtained in the X and Y directions, as this leads to the most demanding condition for the group of buildings and since it is not possible to predict the main direction of seismic action at priori.

4.1.2. Local behaviour

  • Following the discussion in (Simões et al., 2018; see §5.3.1), two possible scenarios were identified for the local seismic behaviour of the buildings related to the response of: 1) the last floor of the building and 2) the parapet.
  • 1 – overturning of the central pier; Mech.
  • Figure 18 plots the capacity curves obtained from the models set to perform the full factorial analyses and the models defined by the median properties of the aleatory variables.
  • The two possible scenarios were assumed as epistemic uncertainties and treated by the logic-tree approach.
  • Figure 19 provides the resulting fragility curves for both scenarios and compares the curves obtained by considering seismic action type 1 and type 2.

4.2. Comparison between classes of buildings

  • This section compares the individual fragility curves for the classes of buildings, after the combination between global and local seismic behaviour, as described in (Simões et al., 2019a; see §6.3).
  • First, the global behaviour in the Y direction of the structure is combined with the local behaviour.
  • The corresponding values are presented in Table 1 and Table 2, respectively, for seismic action type 1 and type 2.
  • Due to the different contributions, the resulting fragility curves are not a lognormal cumulative distribution function.
  • It is also observed that, in general, higher dispersion is obtained for PL1 and PL2 while the dispersion for PL3 and PL4 is similar across the different classes of buildings.

5. Fragility curves for the typical URM tall buildings

  • This section addresses the derivation of the fragility functions for the typical URM tall buildings by considering the contribution of the different classes of buildings identified.
  • These fragility curves take into account the different sources of uncertainties that influence the seismic performance of the buildings, providing the overall assessment of the seismic vulnerability of this class of buildings at territorial scale.
  • The final fragility curves are obtained by adding the fragility curves of the different classes of building as a function of their probability (wj).
  • This is defined in an approximated way by the application of Equation (2) and Equation (3):.

6. Final Remarks

  • The article presents the overall seismic vulnerability assessment of a class of tall URM buildings supported on the definition of different classes of buildings.
  • These features are considered as epistemic uncertainties (related to the incomplete knowledge) and are treated by the logic-tree approach.
  • For each class of buildings, fragility curves are generated taking into account both the in-plane plane and out-of-plane seismic response, following the procedure proposed in a companion article (Simões et al., 2019a).
  • The article also addresses strategies to improve the procedure for the derivation of fragility functions.
  • Preliminary non-linear static analyses indicated that some classes of buildings present a similar global response allowing to reduce the number of classes to be considered in the final and detailed assessment.

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1
Fragility functions for tall URM buildings around early 20
th
century in Lisbon.
Part 2: application to different classes of buildings
Ana G. Simões
1
, Rita Bento
1*
, Sergio Lagomarsino
2
, Serena Cattari
2
, Paulo B. Lourenço
3
1
CERIS, Instituto Superior Técnico, Universidade de Lisboa, Lisbon (Portugal)
2
DICCA, Università Degli Studi di Genova, Genoa (Italy)
3
ISISE, Universidade do Minho, Campus de Azurém, Guimarães (Portugal)
*Corresponding Author: Address: Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal;
Email: rita.bento@tecnico.ulisboa.pt; Tel: +351 218 418 210
Abstract
This article describes the application of the procedure for the derivation of fragility functions presented in the
companion article entitled Fragility functions for tall URM buildings around early 20th century in Lisbon. Part
1: methodology and application at building level. The procedure, based on the execution of non-linear analyses,
was developed to be applied to unreinforced masonry buildings considering both the in-plane and out-of-plane
response. Different sources of uncertainty, both epistemic and aleatory, affecting the behaviour of these
unreinforced masonry buildings are discussed and treated with a probabilistic procedure. The fragility curves
determined for the different classes of buildings are compared and then combined to define the final fragility
curves for these unreinforced masonry buildings. The results put in evidence the high seismic vulnerability of
these buildings and the urgent need for the structural intervention and for the design of retrofitting measures in
order to reduce potential losses due to future earthquakes.
Keywords
Unreinforced Masonry Buildings; Seismic Vulnerability; Fragility Functions; Non-Linear Static Analysis; Non-
Linear Kinematic Analysis; Non-Linear Dynamic Analysis
2
1. Introduction
Seismic vulnerability addresses the susceptibility to suffer damage or loss due to an earthquake. It can be defined
using fragility functions, providing the probability of reaching or exceeding a specified limit state as a function of
a selected seismic intensity measure. The main objective of this work is to apply, to different classes of
unreinforced masonry (URM) buildings around early 20th century in Lisbon, the procedure proposed for the
derivation of fragility functions in the companion article (Part 1, Simões et al., 2019a). This procedure is based
on non-linear analyses and takes into account different sources of uncertainties that influence the seismic
performance.
In Simões et al. (2019a) the main steps of the procedure are presented. Basically, it comprehends the generation
of fragility functions for the in-plane and out-of-plane response following different criteria and methods of
analyses. The in-plane response refers to the global (box-type) behaviour controlled by the in-plane capacity of
walls and stiffness of horizontal diaphragms. The out-of-plane response refers to the activation of local
mechanisms, typically consisting on the overturning of parts of the building insufficiently connected to the rest of
the structure. The fragility curves individually obtained for the in-plane and out-of-plane response are after
combined in order to define a single fragility curve. The capacity curves are evaluated, for the in-plane response,
through non-linear static (pushover) analyses, and, for the out-of-plane response, through non-linear kinematic
analyses. The intensity measure that produces the attainment of the limit state is obtained from the application of
the Capacity-Spectrum Method with overdamped spectrum, as proposed in (Lagomarsino and Cattari, 2015). The
method comprises the comparison between the displacement capacity of the structure for a limit state threshold
and the seismic demand, in an acceleration-displacement coordinates system (refer to §2 from Part 1, Simões et
al., 2019a). The final dispersion of the fragility curves results from the contribution of both aleatory and epistemic
uncertainties on the structural capacity and the aleatory uncertainty on the seismic demand.
This article addresses the application of the procedure proposed in Simões et al. (2019a) to one of the most
vulnerable unreinforced masonry buildings constructed in the early 20
th
century in Lisbon, examining a typical
prototype building with five storeys high. Considering that these buildings are located in the middle or in the end
of a row of buildings, a group of three equal buildings is considered to evaluate the effect of the aggregate in the
seismic behaviour.
Then, for this type of buildings, proper classes are defined (Simões et al., 2018; Simões, 2018) as a function of
their similarity in structural details, material and seismic response. The features that mostly affect the seismic
performance are identified depending on the available data and characteristics of the masonry buildings under
investigation. In this work, these features are considered as epistemic uncertainties (related to the incomplete
knowledge) and are treated by the logic-tree approach. Moreover, effective strategies to reduce the number of
buildings classes and, thus limit the computational effort required in executing the non-linear static analyses, are
discussed. Finally, the fragility curves determined for the different classes of buildings, selected as the “minima”
required to obtain a reliable description of the behaviour of the structural typology under examination, are
compared and the final fragility curves for these unreinforced masonry buildings are provided.
2. The URM “gaioleiro” buildings
2.1. Identification of classes of buildings
The development of vulnerability models at territorial scale firstly requires the identification of classes of
buildings. This is supported on the idea that buildings with similar architectural and structural features and located
in similar geotechnical conditions are expected to have a similar seismic performance. Thus, the features that
mostly affect the seismic performance must be singled out depending on the characteristics of the building stock.
As mentioned in §1, different classes of buildings have been identified in previous works for the URM “gaioleiro”
buildings in Lisbon (Simões et a.l, 2018; Simões, 2018). The different classes of buildings account for geometry
variations at the ground floor level, constructive details and materials attributed to the different walls. All these
features are considered as epistemic uncertainties (related to the incomplete knowledge) and are treated by the
logic-tree approach. A subjective probability is attributed to each feature based on the information available in the
literature and on a detailed survey to a block of URM buildings in “Avenidas Novas” (Simões et al., 2016). This
aims to define different prototypes representative of the URM buildings constructed in Lisbon in the early 20
th
century, starting from the prototype building defined in (Simões et al., 2019a).
Geometry: ground floor configuration
The ground floor level may be used as housing or as shop, implying a different layout of openings as exemplified
in Figure 1. Taking into account the reference block of buildings that has been surveyed, the ground floor of 33%
of the buildings is used as housing and 67% is used for shop. The regular floors are used as housing in both cases.
3
(a) (b) (c) (d)
Figure 1 – Prototype building as housing (a) plan view and (b) street view, and as shop (c) plan view and (d)
street view
Constructive Details
The typical building is located in the middle or in the end of a row of buildings; no isolated configuration can
occur. According to the 1867 law (Código Civil, Decreto 01/07/1867, cited in (Appleton, 2005)) the side walls
could be shared between adjacent buildings. The decision may depend on the time of construction and on the
dimension of the lot. In the reference block of buildings, 33% of the buildings have shared side walls and 67%
have independent side walls. In order to take into account the effect of the adjacent buildings (i) and the possibility
that the side walls are shared or independent between buildings (ii), it is proposed to replicate the prototype
building and define a group of three buildings as case of study, as shown in Figure 2.
(a) (b) (c)
Figure 2 – URM buildings: (a) view from the block of buildings in “Avenidas Novas”, (b) group of buildings
with shared side walls and (c) group of buildings with independent side walls
Materials
One of the main features affecting the seismic performance of URM buildings are the materials and mechanical
properties of the materials used in the construction of the walls. During this period of construction in Lisbon, the
URM buildings may include rubble stone masonry walls, clay brick (solid or hollow units) masonry walls and
timber walls. However, there is few information in the buildings’ process (in Arquivo Municipal de Lisboa, AML,
http://arquivomunicipal.cm-lisboa.pt/) about the material employed in the different walls. Due to this limitation,
all possibilities are examined and a subjective probability is attributed in order to quantify the representativeness
of the option within the building stock.
Exterior walls were constructed in rubble stone masonry on the street and rear façade walls and clay brick masonry
on the side walls. Brick units may be: i) solid in all floors or ii) solid in the lower floors and hollow in the upper
floors. The Building Regulation from 1930 (RGEUL, 1930) recommended to use hollow bricks only in the last
two floors of the side walls. Considering that this regulation was published in the end of the construction age of
these buildings, a lower probability is attributed to this option. Therefore, it may be assumed that the side walls
of 70% of the buildings are made of solid clay brick masonry, while the remaining 30% are made of solid clay
brick masonry in the first three floors and hollow clay brick masonry in the last two floors.
In what concerns interior walls, it is estimated that the main walls are made of solid clay brick masonry in 50%
of the buildings while they are made of solid clay brick masonry in the ground floor and hollow clay brick masonry
in the other floors in 50% of the buildings. However, there are records of buildings in which most of the interior
walls are made with timber structure. In order to consider also this option, it is proposed to assume that in 20% of
the buildings the main interior walls are made of timber (but limiting them to the last floor of the buildings, as
indicated in (RGEUL, 1930)) and to reduce the previous probabilities associated to clay brick masonry to 40/40.
As to the partition walls, it is estimated that in 50% of the buildings these are made of hollow clay brick masonry,
while in 50% of the buildings these are made of timber structure.
X
Y
4
2.2. Logic-Tree Approach: epistemic uncertainties
From the data illustrated in §2.1, the main variations identified for the tall URM “gaioleiro” buildings are
summarized as:
1. Ground floor configuration: use of the building as i) housing (H) or as ii) shop (S).
2. Side walls solution: i) side walls shared between adjacent buildings (S) or ii) independent (I).
3. Side walls material: i) solid clay brick masonry (S) or ii) solid clay brick masonry in the first two floors
and hollow clay brick masonry in the last three floors (SH).
4. Main interior walls material: i) solid clay brick masonry (S), ii) solid clay brick masonry in the first two
floors and hollow clay brick masonry in the last three floors (SH), or iii) solid clay brick masonry in the
first two floors, hollow clay brick masonry in the medium floors and timber walls on the last floor (T).
5. Partition walls material: i) hollow brick masonry (H) or ii) timber walls (T).
These variations are considered as epistemic uncertainties and are organized in a logic-tree. This allows to define
different classes representative of the URM buildings constructed in Lisbon in the early 20
th
century. Each node
of the tree represents a specific feature of the buildings and each branch represents an alternative option associated
to that feature. Finally, the end of a branch of the tree represents a class of buildings with specific features,
identified by an acronym (starting from the capital letters in the numbers above). The probability attributed to the
class of buildings is determined by multiplying the probability of all the component branches of the tree. The
logic-tree proposed for the tall URM “gaioleiro” buildings is presented in Figure 3, comprising a total of 32
building classes (identified as group A).
5
Figure 3 – Logic-tree for the tall URM “gaioleiro” buildings
Citations
More filters
Journal Article•DOI•
Seismic fragility assessment of existing masonry buildings in aggregate

[...]

Michele Angiolilli1, Sergio Lagomarsino1, Serena Cattari1, Stefania Degli Abbati1•
University of Genoa1
15 Nov 2021-Engineering Structures
TL;DR: In this article, the authors describe the derivation of fragility curves useful for the seismic risk analyses of existing unreinforced masonry buildings inserted in aggregate, which consists of three adjacent structural units that may mutually interact during seismic events.

26 citations

Journal Article•DOI•
Seismic assessment of nineteenth and twentieth centuries URM buildings in Lisbon: structural features and derivation of fragility curves

[...]

Ana Simões1, Rita Bento1, Sergio Lagomarsino2, Serena Cattari2, Paulo B. Lourenço3 •
Instituto Superior Técnico1, University of Genoa2, University of Minho3
01 Jan 2020-Bulletin of Earthquake Engineering
TL;DR: In this article, the seismic vulnerability assessment of a typology of unreinforced masonry buildings constructed in Lisbon between the nineteenth and the twentieth centuries is presented. And the structural intervention on these buildings is urgent in order to reduce losses due to future earthquakes.
Abstract: The article addresses the seismic vulnerability assessment of a typology of unreinforced masonry buildings constructed in Lisbon between the nineteenth and the twentieth centuries. The main architectural and structural features of these buildings are presented. This supported the identification of the main uncertainties affecting their seismic performance and the definition of classes of buildings representative of the typology. The seismic assessment includes the generation of fragility curves that combine the in-plane and out-of-plane response following different criteria and methods of analyses. The results put in evidence the seismic vulnerability of this class of buildings. Considering the earthquake-resistant code for Lisbon with a return period of 475 years, about 50% probability of having heavy damage and about 30% probability of collapse were estimated. The structural intervention on these buildings is urgent in order to reduce losses due to future earthquakes. Further studies for the assessment of similar buildings in Lisbon and elsewhere can be developed using the adopted procedure.

18 citations

Journal Article•DOI•
Fragility Functions for Tall URM Buildings around Early 20th Century in Lisbon. Part 1: Methodology and Application at Building Level

[...]

Ana Simões1, Rita Bento1, Sergio Lagomarsino2, Serena Cattari2, Paulo B. Lourenço3 •
Instituto Superior Técnico1, University of Genoa2, University of Minho3
04 Mar 2021-International Journal of Architectural Heritage
TL;DR: In this paper, a procedure for the derivation of fragility functions for unreinforced masonry buildings considering the in-plane and out-of-plane response is proposed.
Abstract: The article proposes a procedure for the derivation of fragility functions for unreinforced masonry buildings considering the in-plane and out-of-plane response. Different approaches are considered for the generation of the corresponding fragility functions and for the evaluation of the propagation of uncertainties. The contributions for the dispersion of the fragility functions account for the variability in the definition of the capacity, the aleatory uncertainty in the definition of the seismic demand and the aleatory uncertainty in the definition of the modified/floor response spectrum, when the local mechanisms are located in the upper level of the building. In the end, the individual fragility curves are properly combined in order to define a single fragility curve for the class of buildings. As a case study, the procedure is applied to the assessment of one of the most vulnerable unreinforced masonry buildings constructed in the early 20th century in Lisbon, considering a typical prototype ...

10 citations

Cites background or methods or result from "Fragility Functions for Tall URM Bu..."

  • ...These modes correspond to the translation of the structure in the direction of the out-of-plane mechanism (Simões et al. 2019b)....

    [...]

  • ...In a companion article (Simões et al. 2019a) the procedure is applied to different classes of buildings aiming at the overall assessment of the seismic vulnerability of this class of buildings in Lisbon....

    [...]

  • ...The counterpart article (Simões et al. 2019a) presents the main features of the classes of buildings, compares their seismic behaviour and provides the final fragility curves for this class of tall URM buildings....

    [...]

  • ...For more information on the local behaviour of the class of buildings check §4.2 from (Simões et al. 2019a)....

    [...]

  • ...The companion article (Simões et al. 2019a) deals with the derivation of the fragility functions for different classes of buildings....

    [...]

Journal Article•DOI•
Fragility functions for local failure mechanisms in unreinforced masonry buildings: a typological study in Ferrara, Italy

[...]

Marco Nale1, Fabio Minghini1, Andrea Chiozzi1, Antonio Tralli1•
University of Ferrara1
01 Nov 2021-Bulletin of Earthquake Engineering
TL;DR: In this article, the authors derived typological fragility functions for unreinforced masonry walls considering OOP local failure mechanisms, where a wall is assumed either as a single rigid body undergoing simple one-sided rocking or a system of two coupled rigid bodies rocking along their common edge.
Abstract: Unreinforced masonry buildings undergoing seismic actions often exhibit local failure mechanisms which represent a serious life-safety hazard, as recent strong earthquakes have shown. Compared to new buildings, older unreinforced masonry buildings are more vulnerable, not only because they have been designed without or with limited seismic loading requirements, but also because horizontal structures and connections amid the walls are not always effective. Also, Out-Of-Plane (OOP) mechanisms can be caused by significant slenderness of the walls even if connections are effective. The purpose of this paper is to derive typological fragility functions for unreinforced masonry walls considering OOP local failure mechanisms. In the case of slender walls with good material properties, the OOP response can be modeled with reference to an assembly of rigid bodies undergoing rocking motion. In particular, depending on its configuration, a wall is assumed either as a single rigid body undergoing simple one-sided rocking or a system of two coupled rigid bodies rocking along their common edge. A set of 44 ground motions from earthquake events occurred from 1972 to 2017 in Italy is used in this study. The likelihood of collapse is calculated via Multiple Stripe Analysis (MSA) from a given wall undergoing a specific ground motion. Then, the single fragility functions are suitably combined to define a typological fragility function for a class of buildings. The procedure is applied to a historical aggregate in the city center of Ferrara (Italy) as a case study. The fragility functions developed in this research can be a helpful tool for assessing seismic damage and economic losses in unreinforced masonry buildings on a regional scale.

6 citations

Journal Article•DOI•
Probabilistic seismic assessment of a high-rise URM building

[...]

Jorge Arturo Avila-Haro1, Ramón González-Drigo1, Yeudy F. Vargas-Alzate1, Lluis Pujades1, Alex H. Barbat1 •
Polytechnic University of Catalonia1
01 Jan 2022-Journal of building engineering
TL;DR: In this paper, a detailed quantification of the uncertainties related to the mechanical properties of this construction material is conducted, and the influence of this variability on the seismic performance of a representative building model of the Eixample district in Barcelona, Spain, is analyzed.
Abstract: According to the Sendai Framework for Disaster Risk Reduction (2015–2030), disasters have demonstrated that the recovery, rehabilitation and reconstruction phase, which needs to be prepared ahead of a disaster, is a critical opportunity to “Build Back Better”, integrating disaster risk reduction into development measures. In this respect, a significant number of structures, that constitute several European urban nuclei, belong to old constructive typologies, which were designed and built without any consideration for the seismic hazard. One of the most used typologies exhibiting this shortcoming is unreinforced masonry (URM). Therefore, an important step towards increasing resilience of European cities is to deeply understand the seismic behavior of this frequent typology. In order to do so properly, detailed probabilistic nonlinear building models should be developed. However, including the uncertainties associated with this typology is challenging due to the heterogeneity of the different manufacturing techniques, executed under primitive industrial standards, and to the construction techniques, which are dependent on regional uses and criteria in a pre-code scenario. The object of this research is twofold. First, a detailed quantification of the uncertainties related to the mechanical properties of this construction material is conducted. Then, the influence of this variability on the seismic performance of a representative building model of the Eixample district in Barcelona, Spain, is analysed. This building typology represents 72% of the building stock in this district with an average age of 90 years, which means that the construction practice, at that time, was only regulated by early council guidelines that are considered pre-code rules. Specifically, the probabilistic approach is illustrated with a case study performed on an existing seven-story (high-rise) URM. A detailed numerical model of this structure has been developed and randomized taking into account the variability of the material properties. Accordingly, 1000 models were generated and analysed by considering as input different sets of material random variables. The compressive strength, Young modulus, shear modulus and shear strength are chosen and modelled to encompass the material uncertainties. The seismic response of each variant (i.e. selected set of mechanical properties) is obtained through a simplified non-linear static procedure aiming to compare and categorize the influence of the probabilistic input on the seismic performance of the building. Results are presented in terms of correlations between damage parameters and material properties. The analysis carried out shows that the variability in the material properties generates significant uncertainties in the seismic response of URM buildings, leading to over or underestimate expected damage when compared with results based on approaches that do not consider the probabilistic nature of the problem.

5 citations

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01 Jan 1995
TL;DR: In this paper, the authors present a single-degree-of-freedom (SDF) system, which is composed of a mass-spring-damper system and a non-viscous Damping Free Vibration (NFV) system.
Abstract: I. SINGLE-DEGREE-OF-FREEDOM SYSTEMS. 1. Equations of Motion, Problem Statement, and Solution Methods. Simple Structures. Single-Degree-of-Freedom System. Force-Displacement Relation. Damping Force. Equation of Motion: External Force. Mass-Spring-Damper System. Equation of Motion: Earthquake Excitation. Problem Statement and Element Forces. Combining Static and Dynamic Responses. Methods of Solution of the Differential Equation. Study of SDF Systems: Organization. Appendix 1: Stiffness Coefficients for a Flexural Element. 2. Free Vibration. Undamped Free Vibration. Viscously Damped Free Vibration. Energy in Free Vibration. Coulomb-Damped Free Vibration. 3. Response to Harmonic and Periodic Excitations. Viscously Damped Systems: Basic Results. Harmonic Vibration of Undamped Systems. Harmonic Vibration with Viscous Damping. Viscously Damped Systems: Applications. Response to Vibration Generator. Natural Frequency and Damping from Harmonic Tests. Force Transmission and Vibration Isolation. Response to Ground Motion and Vibration Isolation. Vibration-Measuring Instruments. Energy Dissipated in Viscous Damping. Equivalent Viscous Damping. Systems with Nonviscous Damping. Harmonic Vibration with Rate-Independent Damping. Harmonic Vibration with Coulomb Friction. Response to Periodic Excitation. Fourier Series Representation. Response to Periodic Force. Appendix 3: Four-Way Logarithmic Graph Paper. 4. Response to Arbitrary, Step, and Pulse Excitations.Response to Arbitrarily Time-Varying Forces. Response to Unit Impulse. Response to Arbitrary Force. Response to Step and Ramp Forces. Step Force. Ramp or Linearly Increasing Force. Step Force with Finite Rise Time. Response to Pulse Excitations. Solution Methods. Rectangular Pulse Force. Half-Cycle Sine Pulse Force. Symmetrical Triangular Pulse Force. Effects of Pulse Shape and Approximate Analysis for Short Pulses. Effects of Viscous Damping. Response to Ground Motion. 5. Numerical Evaluation of Dynamic Response. Time-Stepping Methods. Methods Based on Interpolation of Excitation. Central Difference Method. Newmark's Method. Stability and Computational Error. Analysis of Nonlinear Response: Central Difference Method. Analysis of Nonlinear Response: Newmark's Method. 6. Earthquake Response of Linear Systems. Earthquake Excitation. Equation of Motion. Response Quantities. Response History. Response Spectrum Concept. Deformation, Pseudo-Velocity, and Pseudo-Acceleration Response Spectra. Peak Structural Response from the Response Spectrum. Response Spectrum Characteristics. Elastic Design Spectrum. Comparison of Design ad Response Spectra. Distinction between Design and Response Spectra. Velocity and Acceleration Response Spectra. Appendix 6: El Centro, 1940 Ground Motion. 7. Earthquake Response of Inelastic Systems. Force-Deformation Relations. Normalized Yield Strength, Yield Strength Reduction Factor, and Ductility Factor. Equation of Motion and Controlling Parameters. Effects of Yielding. Response Spectrum for Yield Deformation and Yield Strength. Yield Strength and Deformation from the Response Spectrum. Yield Strength-Ductility Relation. Relative Effects of Yielding and Damping. Dissipated Energy. Energy Dissipation Devices. Inelastic Design Spectrum. Applications of the Design Spectrum. Comparison of Design and Response Spectra. 8. Generalized Single-Degree-of-Freedom Systems. Generalized SDF Systems. Rigid-Body Assemblages. Systems with Distributed Mass and Elasticity. Lumped-Mass System: Shear Building. Natural Vibration Frequency by Rayleigh's Method. Selection of Shape Function. Appendix 8: Inertia Forces for Rigid Bodies. II. MULTI-DEGREE-OF-FREEDOM SYSTEMS. 9. Equations of Motion, Problem Statement, and Solution Methods. Simple System: Two-Story Shear Building. General Approach for Linear Systems. Static Condensation. Planar or Symmetric-Plan Systems: Ground Motion. Unsymmetric-Plan Building: Ground Motion. Symmetric-Plan Buildings: Torsional Excitation. Multiple Support Excitation. Inelastic Systems. Problem Statement. Element Forces. Methods for Solving the Equations of Motion: Overview. 10. Free Vibration. Natural Vibration Frequencies and Modes. Systems without Damping. Natural Vibration Frequencies and Modes. Modal and Spectral Matrices. Orthogonality of Modes. Interpretation of Modal Orthogonality. Normalization of Modes. Modal Expansion of Displacements. Free Vibration Response. Solution of Free Vibration Equations: Undamped Systems. Free Vibration of Systems with Damping. Solution of Free Vibration Equations: Classically Damped Systems. Computation of Vibration Properties. Solution Methods for the Eigenvalue Problem. Rayleigh's Quotient. Inverse Vector Iteration Method. Vector Iteration with Shifts: Preferred Procedure. Transformation of kA A = ...w2mA A to the Standard Form. 11. Damping in Structures.Experimental Data and Recommended Modal Damping Ratios. Vibration Properties of Millikan Library Building. Estimating Modal Damping Ratios. Construction of Damping Matrix. Damping Matrix. Classical Damping Matrix. Nonclassical Damping Matrix. 12. Dynamic Analysis and Response of Linear Systems.Two-Degree-of-Freedom Systems. Analysis of Two-DOF Systems without Damping. Vibration Absorber or Tuned Mass Damper. Modal Analysis. Modal Equations for Undamped Systems. Modal Equations for Damped Systems. Displacement Response. Element Forces. Modal Analysis: Summary. Modal Response Contributions. Modal Expansion of Excitation Vector p (t) = s p(T). Modal Analysis for p (t) = s p(T). Modal Contribution Factors. Modal Responses and Required Number of Modes. Special Analysis Procedures. Static Correction Method. Mode Acceleration Superposition Method. Analysis of Nonclassically Damped Systems. 13. Earthquake Analysis of Linear Systems.Response History Analysis. Modal Analysis. Multistory Buildings with Symmetric Plan. Multistory Buildings with Unsymmetric Plan. Torsional Response of Symmetric-Plan Buildings. Response Analysis for Multiple Support Excitation. Structural Idealization and Earthquake Response. Response Spectrum Analysis. Peak Response from Earthquake Response Spectrum. Multistory Buildings with Symmetric Plan. Multistory Buildings with Unsymmetric Plan. 14. Reduction of Degrees of Freedom. Kinematic Constraints. Mass Lumping in Selected DOFs. Rayleigh-Ritz Method. Selection of Ritz Vectors. Dynamic Analysis Using Ritz Vectors. 15. Numerical Evaluation of Dynamic Response. Time-Stepping Methods. Analysis of Linear Systems with Nonclassical Damping. Analysis of Nonlinear Systems. 16. Systems with Distributed Mass and Elasticity. Equation of Undamped Motion: Applied Forces. Equation of Undamped Motion: Support Excitation. Natural Vibration Frequencies and Modes. Modal Orthogonality. Modal Analysis of Forced Dynamic Response. Earthquake Response History Analysis. Earthquake Response Spectrum Analysis. Difficulty in Analyzing Practical Systems. 17. Introduction to the Finite Element Method.Rayleigh-Ritz Method. Formulation Using Conservation of Energy. Formulation Using Virtual Work. Disadvantages of Rayleigh-Ritz Method. Finite Element Method. Finite Element Approximation. Analysis Procedure. Element Degrees of Freedom and Interpolation Function. Element Stiffness Matrix. Element Mass Matrix. Element (Applied) Force Vector. Comparison of Finite Element and Exact Solutions. Dynamic Analysis of Structural Continua. III. EARTHQUAKE RESPONSE AND DESIGN OF MULTISTORY BUILDINGS. 18. Earthquake Response of Linearly Elastic Buildings. Systems Analyzed, Design Spectrum, and Response Quantities. Influence of T 1 and r on Response. Modal Contribution Factors. Influence of T 1 on Higher-Mode Response. Influence of r on Higher-Mode Response. Heightwise Variation of Higher-Mode Response. How Many Modes to Include. 19. Earthquake Response of Inelastic Buildings. Allowable Ductility and Ductility Demand. Buildings with "Weak" or "Soft" First Story. Buildings Designed for Code Force Distribution. Limited Scope. Appendix 19: Properties of Multistory Buildings. 20. Earthquake Dynamics of Base-Isolated Buildings. Isolation Systems. Base-Isolated One-Story Buildings. Effectiveness of Base Isolation. Base-Isolated Multistory Buildings. Applications of Base Isolation. 21. Structural Dynamics in Building Codes. Building Codes and Structural Dynamics. International Building Code (United States), 2000. National Building Code of Canada, 1995. Mexico Federal District Code, 1993. Eurocode 8. Structural Dynamics in Building Codes. Evaluation of Building Codes. Base Shear. Story Shears and Equivalent Static Forces. Overturning Moments. Concluding Remarks. Appendix A: Frequency Domain Method of Response Analysis.Appendix B: Notation.Appendix C: Answers to Selected Problems.Index.

4,812 citations

"Fragility Functions for Tall URM Bu..." refers background in this paper

  • ...Here, it is noted that the response of the building in terms of dissipated energy is a composition of inelastic behaviour and viscous damping, Chopra (2016), meaning that higher damping of the response is obtained....

    [...]

Journal Article•DOI•
Simulation and the Monte Carlo Method.

[...]

Thomas M. F. O'Donovan, Reuven Y. Rubinstein
01 Mar 1983-Biometrics

1,897 citations

"Fragility Functions for Tall URM Bu..." refers methods in this paper

  • ...TheMonte Carlo Method (Rubinstein 2011) is used to define possible outcome values for each aleatory variables....

    [...]

Journal Article•DOI•
TREMURI program: An equivalent frame model for the nonlinear seismic analysis of masonry buildings

[...]

Sergio Lagomarsino1, Andrea Penna2, Alessandro Galasco2, Serena Cattari1•
University of Genoa1, University of Pavia2
01 Nov 2013-Engineering Structures
TL;DR: In this paper, the authors present the solutions adopted for the implementation of the equivalent frame model in the TREMURI program for the nonlinear seismic analysis of masonry buildings, which is also expressly recommended in several national and international codes.

475 citations

"Fragility Functions for Tall URM Bu..." refers background or methods in this paper

  • ...The analyses are performed in Tremuri program (Lagomarsino et al. 2013) considering the twomain directions of the group of buildings (X— parallel to the façade walls and Y—parallel to the side ) c ( ) b ( ) a ( X Y...

    [...]

  • ...Non-linear dynamic analyses are performed with Tremuri program (Lagomarsino et al. 2013) with the objective of checking if the load distributions considered in the non-linear static (pushover) analyses are able to capture the real capacity of the building....

    [...]

  • ...Tremuri program (Lagomarsino et al. 2013) with the objective of checking if the load distributions considered in the non-linear static (pushover) analyses are...

    [...]

  • ...The analyses are performed in Tremuri program (Lagomarsino et al. 2013) considering the twomain directions of the group of buildings (X— parallel to the façade walls and Y—parallel to the side )c()b()a( X Y Figure 2....

    [...]

Journal Article•DOI•
1,1‐Dicyano‐4‐[4‐(diethylamino)phenyl]buta‐1,3‐dienes: Structure–Property Relationships

[...]

Francesca Tancini1, Yi Lin Wu1, W. Bernd Schweizer1, Jean Paul Gisselbrecht2, Corinne Boudon2, Peter D. Jarowski3, Marten T. Beels4, Ivan Biaggio4, François Diederich1 •
ETH Zurich1, University of Strasbourg2, University of Surrey3, Lehigh University4
01 May 2012-European Journal of Organic Chemistry
TL;DR: In this paper, a series of 1,1- dicyano-4-[4-(diethylamino)phenyl]buta-1,3-dienes were synthesized and the number and position of additional CN substituents along the 1, 1-dicyanobuta 1,3dienyl fragment was systematically varied.

241 citations

Monograph•DOI•
Simulation and Monte Carlo

[...]

John Dagpunar
26 Jan 2007

85 citations

"Fragility Functions for Tall URM Bu..." refers methods in this paper

  • ...TheMonte Carlo Method (Rubinstein 2011) is used to define possible outcome values for each aleatory...

    [...]

  • ...TheMonte Carlo Method (Rubinstein 2011) is used to define possible outcome values for each aleatory variables....

    [...]

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Frequently Asked Questions (11)
Q1. What have the authors contributed in "Fragility functions for tall urm buildings around early 20th century in lisbon. part 2: application to different classes of buildings" ?

This article describes the application of the procedure for the derivation of fragility functions presented in the companion article entitled Fragility functions for tall URM buildings around early 20th century in Lisbon. Different sources of uncertainty, both epistemic and aleatory, affecting the behaviour of these unreinforced masonry buildings are discussed and treated with a probabilistic procedure. The results put in evidence the high seismic vulnerability of these buildings and the urgent need for the structural intervention and for the design of retrofitting measures in order to reduce potential losses due to future earthquakes. 

The time-dependent response of the structure is obtained through direct numerical integration of the differential equations of motion of the system, considering both horizontal components of the acceleration spectra acting simultaneously. 

Concerning the global response, the main criticality of the buildings in the direction of the side walls, is related to the insufficient capacity in terms of ductility more than overall strength, while in the direction of the façade walls the opposite occurs. 

it was also concluded that the final dispersion is mainly due to the contribution of the dispersion in the seismic demand (D), due to the large variability of possible ground-motion records. 

The final fragility curves are obtained by adding the fragility curves of the different classes of building as a function of their probability (wj). 

Non-linear dynamic analyses are performed with Tremuri program (Lagomarsino et al. 2013) with the objective of checking if the load distributions considered in the non-linear static (pushover) analyses are able to capture the real capacity of the building. 

The effects of viscous damping are considered by adopting the Rayleigh damping formulation and assuming a viscous damping constant and equal to 3%. 

The probability attributed to the class of buildings is determined by multiplying the probability of all the component branches of the tree. 

In order to take into account the effect of the adjacent buildings (i) and the possibility that the side walls are shared or independent between buildings (ii), it is proposed to replicate the prototype building and define a group of three buildings as case of study, as shown in Figure 2. 

As in the global behaviour, it is observed that the most demanding condition for the structure is obtained with seismic action type 1, defined by lower values of PGA50%. 

One of the main features affecting the seismic performance of URM buildings are the materials and mechanical properties of the materials used in the construction of the walls.