# Innovative Fragility-Based Method for Failure Mechanisms and Damage Extension Analysis of Bridges

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- The failure mode analysis, consisting of a qualitative estimation of the most significant failure modes and related mutual relations, provides useful insights for the selection of the most suitable retrofit intervention.
- The damage extent analysis, consisting of a quantitative assessment of the damage spreading over the system, is useful to perform loss analysis, prioritisation of repair interventions (e.g., within a given infrastructural network), and develop a rational design of retrofit strategies.

## 2. Proposed Methodology for the Fragility Analysis of Bridges

_{k}, with k = 1,…N

_{Ak}, where N

_{Ak}is the number of components that may suffer the type of failure A. Consequently, the failure event A occurs if at least one component experiences failure, i.e., one or more A

_{k}events occur:

_{Gfc}, can be defined (Equation (4)) on the union event U (with U = A∪B∪C…) denoting the global failure event).

#### 2.1. Failure Mode Analysis

_{l}, with l = 1,…N

_{M}, considering all the N

_{M}possible combinations of failure types, such as the occurrence of only one failure mode and the coupling of two or more failure modes. As an example, if three failure modes $A$, $B,$ and $C$ are possible, $U$ is partitioned into seven sets (Figure 1): U\(B∪C), U\(A∪C), U\(A∪B), (A∩C)\B, (B∩C)\A, (A∩B)\C, and (A∩B∩C).

_{l}, i.e.,

#### 2.2. Analysis of the Damage Extent

_{Ak}components to obtain information about the damage extent. This partition is based on the number of damaged elements, e.g., subsets S

_{m}, m = 1…N

_{S}, that may concern failures occurring in 1 element, 2 elements, … all the N

_{Ak}= N

_{S}elements. It is evident that each subset can collect failures of the same family occurring at different locations along the bridge because only the failure number is counted. In this sense, the approach is not able to provide information about the location of damage but only about the damage spreading.

_{S}< N

_{Ak}) is often convenient, considering situations in which failures involve increasing percentages of the elements of the family (e.g., up to 25%, from 25% to 50%, from 50% to 75% and from 75% to 100%). A schematic representation is given in Figure 2, where the failure mode A is studied and S

_{1}, S

_{2},and S

_{3}are used for events involving damage in 25%, 50%, and 75% of the elements of the family.

_{m}can be considered such that,

_{D}being the total number of possible combinations. It follows that the probability of A can also be defined as follows.

#### 2.3. Probabilistic Tools for the Proposed Methodology

_{IM}(im), the latter denoting the Mean Annual Frequency (MAF) of exceeding a given value, im, of the random variable, IM. As for the choice of the IM, the most practical and used ones for the 2D analysis of bridges are the Peak Ground Acceleration (PGA) [15,22] and the spectral acceleration S

_{a}(1.0) at the period of 1.0 s [36,37]. The spectral acceleration Sa (T

_{1}) can be used as well, where T

_{1}is the period of the first mode of vibration; however, this approach may suffer from the generally large difference between the longitudinal and the transverse periods of the bridge. To cope with this issue, an average period, T, between the transversal (1.15 s) and longitudinal vibration (0.67 s) modes is considered in this study. In order to account for the bidirectional nature of the seismic input (X and Y directions), the maximum spectral acceleration component at the period T is selected as the conditioning parameter [38,39], i.e., IM = max{S

_{a,X}(T), S

_{a,Y}(T)}.

_{k}(im) is the indicator function (Ik = 1 if a failure occurs at the k-th time-history analysis for IM = im, I

_{k}= 0 otherwise). Alternatively, starting from the count of the fraction of records, N

_{fi}in each stripe causing failure, a parametric lognormal cumulative distribution, $\mathsf{\Phi}\left(\widehat{\theta},\widehat{\beta}\right),$ can be fitted, which is a reasonable assumption for a large set of cases, whose parameters, $\left\{\widehat{\theta},\widehat{\beta}\right\}$ ($\theta $ expressing the median of the fragility function and β the standard deviation of the natural logarithm of IM), can be evaluated by maximising the logarithm of the likelihood function [43]:

_{IM}as the number of IM levels. Both the empirical and the parametric approaches are used in this work.

#### 2.4. Procedure Implementation

_{A}and F

_{B}) and multiple damage fragility curve F

_{AB}. The procedure can be summarised through the following steps:

- (1)
- Consider family A; for each element (in this case, A
_{1}and A_{2}) a Boolean N_{GM}× N_{IM}matrix is built collecting 1 s for failure cases (i.e., the EDPs exceed a given threshold value) and 0 s otherwise. - (2)
- Compute the union A
_{1}∪A_{2}, which is still a Boolean matrix and characterises the failure states of the whole family A (i.e., failure occurs if at least one component of the family fails). - (3)
- Steps 1 and 2 must be repeated for set B, here supposed to have three sub-elements, hence, three Boolean matrixes are first generated and then the union B
_{1}∪B_{2}∪B_{3}is computed. - (4)
- The resulting union matrixes of A and B can be used individually to evaluate classical single component fragility functions (through either Equation (12) or (13)), i.e., F
_{A}and F_{B}. - (5)
- Additionally, the intersection between the resulting union matrixes of A and B can be computed and used to estimate the multiple damage fragility function F
_{AB}, which adds information about the occurrence of simultaneous failures to the previous fragility types on the considered components.

- (1)
- For each element (B
_{1}, B_{2,}and B_{3}), a Boolean N_{GM}× N_{IM}matrix is built collecting 1 s and 0 s based on the same criterion presented in step 1 of the previous procedure. - (2)
- Compute the matrix sum B
_{1}+ B_{2}+B_{3}, which is no longer Boolean and contains quantitative information on the number of components that failed in family B. - (3)
- Supposing we are interested in characterising the probabilities of having more than two elements at every IM level, a Boolean N
_{GM}× N_{IM}matrix can be built collecting 1 s in the matrix positions where the values are higher than 2 and 0 s where the values are equal or lower than 2. - (4)
- The resulting matrix can be used to build a quantitative damage extent fragility function F
_{B}.

## 3. Case Study

#### 3.1. Seismic Hazard

#### 3.2. Geometrical and Structural Details

_{h}, provided by the above codes, is 7% of the gravity loads. By assuming a concrete grade R

_{ck}300 (cubic characteristic strength equal to 30 MPa) and a steel grade FeB44k (yielding strength equal to 435 MPa), the simulated design requires 59 longitudinal steel rebars of diameter 26 mm for piers and Ø14 mm hoops equally spaced at 25 cm along the whole pier shaft. A longitudinal reinforcement ratio of 1% is assumed for the piers, compatible with the minimum amount required by the code (i.e., 0.6%, [46]). Links are designed assuming the allowable design stresses foreseen by [46] for prestressing steel. As for piers, the confinement effect due to stirrups is considered, assuming confined and unconfined compressive strength and ultimate strains for the cover and core concrete, respectively. The DYWIDAG bars are pre-tensioned at 60% of the steel yielding stress. For sake of completeness, the material constitutive laws are depicted in charts in Figure 8, whereas material properties are reported in Table 2.

^{2}, thickness $t$ = 52 mm, and shear modulus $G$ = 1 MPa are adopted [47].

#### 3.3. Bridge Structural Model

^{3}for the RC and lumped nodal masses are instead used for the pier cap. Gravity and permanent loads are directly derived from distributed and nodal masses and applied before the Time History (TH) nonlinear analysis.

#### 3.4. Demand Parameters and Capacity Limits

_{y}(e.g., piers or DYWIDAG bars yielding or sliding activation for bearings); the attainment of the ultimate capacity of the bridge component, denoted as d

_{u}(e.g., the maximum stroke of the bearings, ultimate strains in the DYWIDAG bars or pier collapse); and an intermediate limit state, d

_{LS}, is introduced to represent an extensive damage condition, related to the Life Safety limit state of the current code [55].

## 4. Results and Discussion

#### 4.1. Failure Modes and Global Fragility Analysis

_{k}contributing to the given failure mode A. For the application of the proposed methodology to the present case study, the following quantities are defined: Mode L-fragility curve characterising the failure of Link (L) bars; Mode P-fragility curve for the Piers (P); Mode B-fragility curve for the Bearings (B); and Mode AP-, Mode AA-, and Mode AT-fragility curves characterising the failure of Abutments for Passive (AP), Active (AA), and Transverse (AT) mechanisms, respectively. In Figure 11, the six failure mode fragility curves (L, P, B, AP, AA, and AT) are compared. Such a representation in the results is quite classical in fragility analysis and allows for the identification of some general trends of the bridge seismic response. For every performance level, the failure modes governing the fragility can be identified based on their relative position within the charts, i.e., ordered from the left (higher fragility) to the right (lower fragility). For the adopted case study, Links are the most vulnerable component while Piers are the less fragile ones. The level of detail of the results is sometimes further reduced by considering a global fragility representation for every performance level, corresponding to the envelope of the single failure mode fragility curves of Figure 11. According to this, the Link’s fragility curves would fully represent the fragility of the whole bridge under investigation. Previous considerations make it evident that in the limit of a classical analysis procedure a large amount of information remains hidden, such as: which link, or group of links, is more vulnerable among the family; how many components (e.g., number of Links) are experiencing failure; and which combination of failure modes (e.g., links and bearings) most govern the failure within a given interval of seismic intensities. To provide such a major level of detail to the bridge response characterisation, the fragility method proposed in Section 2 should be applied, as presented hereafter.

#### 4.2. Multiple Damage Fragility Analysis

_{a}(T) $\le $ 0.5 g), link bars are the only components contributing to the collapse fragility (i.e., global fragility overlaps with links fragility); for higher intensities (0.5 g < S

_{a}(T) $\le $ 1.5 g), bearings start experiencing failure so that the bridge collapse is attained due to a combination of concurring failures on link bars and bearings together; at very high seismic intensities (S

_{a}(T) > 1.5 g), slight participation of the abutments on the global collapse fragility can be observed. The knowledge of the most probable concurring failure mechanisms can help to select the best retrofit intervention and plan monitoring strategies. Moreover, knowing the range of IMs at which specific failure combinations take place, more informed and focused post-earthquake in-situ inspections could be organised depending on the seismic intensity registered at the bridge location.

#### 4.3. Analysis of the Damage Extent

#### 4.3.1. Type 1 Quantitative Fragility Functions

#### 4.3.2. Type 2 Quantitative Fragility Functions

_{a}(T) < 2.0 g, the major contribution to the fragility of piers is that 50% of piers are damaged (blue line); at very high IMs (i.e., S

_{a}(T) > 2.0 g), the condition that all the piers exceed the damage condition is prevalent, with probabilities growing rapidly from about 40% to 80%.

#### 4.3.3. Type 3 Quantitative Fragility Functions

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Analysis of failure modes corresponding to a global failure (event U): an example with three failure modes A, B, and C.

**Figure 2.**Analysis of the damage extent for failure mode A, different subsets of damaged components.

**Figure 5.**(

**a**) IM hazard curve and relevant IM levels; response spectra of the horizontal components of seismic samples at two intensity levels; (

**b**) Tr = 50 years; (

**c**) Tr = 450 years.

**Figure 6.**View of the example bridge. (

**a**) Pre-stressed concrete deck cross-section; (

**b**) scheme of link bar arrangement at the slab level (

**c**) static scheme of bearings for abutments and piers; and (

**d**) longitudinal view of the bridge.

**Figure 7.**(

**a**) Schematic representation of debonded rebar link slab and (

**b**) detail of the joining elements between the decks (link slabs). Source: design project of the road links among SS76 road, A14 highway, Falconara airport, and SS16 road, ANAS, 18.02. 1986 [44].

**Figure 9.**Details of the finite element model of the bridge: (

**a**) deck, link slabs, bearings, and piers; (

**b**) abutments and pounding.

**Figure 11.**Failure mode fragility curves compared at specific performance levels: (

**a**) damage; (

**b**) life safety; and (

**c**) collapse.

**Figure 12.**Multiple damage fragility curves at (

**a**) damage; (

**b**) life safety; and (

**c**) collapse performance levels; envelope global fragility curves depicted by black solid lines.

**Figure 13.**Type 1 quantitative fragility functions for link-slab bars: probabilities of occurrence of damage mechanisms for different percentages of involved links at (

**a**) damage; (

**b**) life safety; and (

**c**) collapse performance levels.

**Figure 14.**Type 2 quantitative fragility functions for piers: probabilities of occurrence of damage mechanisms for different percentages of involved piers at collapse performance levels.

**Figure 15.**Type 3 quantitative fragility curves at (

**a**) slight damage; (

**b**) life safety; and (

**c**) collapse performance levels for bearings: increasing percentages of involved elements (sets of devices).

IM levels | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

S_{a} (T = 0.9) | 0.004 | 0.030 | 0.063 | 0.103 | 0.157 | 0.225 | 0.306 | 0.415 | 0.531 | 0.663 |

IM levels | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

S_{a} (T = 0.9) | 0.802 | 0.976 | 1.140 | 1.316 | 1.529 | 1.693 | 1.823 | 2.134 | 2.470 | 2.816 |

DYWIDAG Steel Bars | Steel Reinforcement | Confined Concrete | Unconfined Concrete | |
---|---|---|---|---|

Elastic modulus (MPa) | 206,000 | 206,000 | 30,000 | 30,000 |

Yielding/Peak strength (MPa) | 950 (σ_{prestress} = 60% yielding) | 435 | 26.75 | 25.00 |

Ultimate strength (MPa) | 1050 | 540 | 18.57 | 3.58 |

Yielding/Peak strain (-) | 0.0046 | 0.00207 | 0.0027 | 0.002 |

Ultimate strain (-) | 0.04 | 0.12000 | 0.0114 | 0.0060 |

Performance Level (PL) | Link Bars ε _{bar} (-) | Piers u _{p} (m) | Bearings u _{b} (m) | Abutment-Passive u _{AB-P} (m) | Abutment-Active u _{AB-A} (m) | Abutment-Transverse u _{AB-T} (m) |
---|---|---|---|---|---|---|

Yielding/Damage (d_{y}) | 0.0046 | 0.077 | 0.080 | 0.037 | 0.00975 | 0.00975 |

Life Safety (d_{SL}) | 3/4 ε_{bar,u} | 3/4 u_{p,u} | 3/4 u_{b,u} | 1.000 | 0.0072 | 0.0072 |

Collapse (d_{u}) | 0.0400 | 0.234 | 0.200 | 1.000 | 1.000 | 1.000 |

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**MDPI and ACS Style**

Minnucci, L.; Scozzese, F.; Carbonari, S.; Gara, F.; Dall’Asta, A.
Innovative Fragility-Based Method for Failure Mechanisms and Damage Extension Analysis of Bridges. *Infrastructures* **2022**, *7*, 122.
https://doi.org/10.3390/infrastructures7090122

**AMA Style**

Minnucci L, Scozzese F, Carbonari S, Gara F, Dall’Asta A.
Innovative Fragility-Based Method for Failure Mechanisms and Damage Extension Analysis of Bridges. *Infrastructures*. 2022; 7(9):122.
https://doi.org/10.3390/infrastructures7090122

**Chicago/Turabian Style**

Minnucci, Lucia, Fabrizio Scozzese, Sandro Carbonari, Fabrizio Gara, and Andrea Dall’Asta.
2022. "Innovative Fragility-Based Method for Failure Mechanisms and Damage Extension Analysis of Bridges" *Infrastructures* 7, no. 9: 122.
https://doi.org/10.3390/infrastructures7090122