PSI - Issue 41

Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2022) 000–000

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ScienceDirect

Procedia Structural Integrity 41 (2022) 1–2

© 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the MedFract2Guest Editors. © 2022 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the MedFract2Guest Editors. Keywords: Preface, Fracture, Structural Integrity 1. Preface At the end of February 2020, the Greek Society of Experimental Mechanics of Materials (GSEMM) and the Italian Group on Fracture (IGF) , respectively, the Greek and the Italian branches of the European Structural Integrity Society (ESIS), organized in Athens the “ 1st Mediterranean Conference on Fracture and Structural Integrity, MedFract1 ” ( MedFract1 ). It was a memorable event, as it was first event in blended version we ever organized, with some participants that were already locked down due to COVID-19 pandemia. After the MedFract1 , for almost two years, many events were organized in remote version by the national groups and by ESIS. At the end of 2021, the Algerian Group of Fracture Mechanics and Energy (AGFME) , the Greek Society of Experimental Mechanics of Materials (GSEMM) , the Italian Group of Fracture (IGF) and the Sociedad Espanola de Integridad Estructural - Grupo Espanol de Fractura (SEIE-GEF) decided to organize the second edition of the MedFract, the “ 2nd Mediterranean Conference on Fracture and Structural Integrity, MedFract2 ”. Considering that the COVID19 problems were still present, the event was organized in blended version, in Catania and on web. The event was undoubtefully a success. In the first edition there were about 60 participants from 15 Thi 2nd Mediterranean Conference on Fracture and Structural Integrity Preface Filippo Berto a , Vittorio Di Cocco b , Francisco Galvez c , Stavros Kourkoulis d , Francesco Iacoviello b *, Mohammed Hadj Meliani e a Norwegian University of Science and Technology, Norway b Department of Civil and Mechanical Engineering, University of Cassino and Southern Lazio, Cassino, Italy c Universidad Politécnica de Madrid, Spain Fi e d National Technical University of Athens, Greece e Université Hassiba benbouali de Chlef, Algeria

* Corresponding author. Tel.: +39 0776 299 681. E-mail address: iacoviello@unicas.it

2452-3216 © 2022 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0 ) Peer-review under responsibility of the MedFract2Guest Editors.

2452-3216 © 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the MedFract2Guest Editors. 10.1016/j.prostr.2022.05.001

Filippo Berto et al. / Procedia Structural Integrity 41 (2022) 1–2

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Author name / Structural Integrity Procedia 00 (2019) 000–000

different countries. In this second edition, there were more than 130 participants from 24 different countries … and many of these countries were not even Mediterranean countries! According to the IGF “tradition”, almost all the presentations were videorecorded, published on the IGF YouTube channel and collected in a Virtual volume that is available in the ESIS PH website at the following address: https://www.esis-ph.eu/index.php/eph/catalog/book/23 This volume collects the papers of many presentations, both in presence and remote. The number and the quality of the papers confirm that the fracture and structural integrity community is really active. The Editors would like to express their sincere thanks to the Authors who contributed to this volume and to all the chairmpersons who helped the organization fo the event!

ScienceDirect Structural Integrity Procedia 00 (2022) 000–000 Structural Integrity Procedia 00 (2022) 000–000 Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceD rect Available online at www.sciencedirect.com ScienceDirect

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Procedia Structural Integrity 41 (2022) 618–630

© 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the MedFract2Guest Editors. Abstract In this work, the effects of damage on the mechanical and vibrational response of reinforced concrete structures is investigated. The proposed model is based on a cohesive interface approach able to simulate the diffuse cracking behavior typical of reinforced concrete structures, in conjunction with an embedded truss model, able to simulate interaction phenomena between steel reinforcement and surrounding concrete. The mathematical model is developed to describe in a realistic way the crack pattern and its evolution and to determine the main factors that can influence the static and dynamic response of damaged reinforced concrete structures. Their static and dynamic properties were evaluated for increasing levels of damage after the removal of the load, starting from the load level at the first crack nucleation to the load level of incipient collapse. In order to verify the accuracy and reliability of the proposed computational model, the numerical results, obtained in terms of variations of the vibrational characteristics of the system will also be compared with experimental data reported in literature. Results show the applicability and reliability of damage identification procedures based on both mathematical models and experimental data for structural systems such as reinforced concrete beams common in girder bridges. © 2022 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the MedFract2Guest Editors. Keywords: Structural Health Monitoring; Reinforced Concrete Beams; Simulation-Based Damage Detection; Modal Properties; Diffuse Cracking; Cohesive Fracture. 2nd Mediterranean Conference on Fracture and Structural Integrity A Cohesive fracture approach for the nonlinear analysis of load induced degradation of vibration characteristics in RC beams Andrea Pranno a , Fabrizio Greco a, *, Paolo Lonetti a , Daniele Gaetano a , Claudio Le Piane b , Umberto De Maio a a Department of Civil Engineering, University of Calabria, Via P. Bucci Cubo 39B, Rende 87036, Italy b Provincial Administratio of Cosenza, Corso Telesio 17, 87100 Cosenza, Italy Abstract In this work, the effects of damage on the mechanical and vibrational response of reinforced concrete structures is i vestigated. T e proposed model is bas d on a cohesive inte face pproach able to simulate the diffuse racking behav or typical of reinforced concrete tructures, in conju ction with an embedded truss model, able to simulate interaction henomena betw en steel r infor em nt and surrounding concret . The mathematical mode is developed to des ribe in a realistic way the crack pattern and its evolutio a d to det rmine the main fa tors that can influence he static and dyn mic response of damaged rei forced c ncrete structur s. Their static d dynamic properties wer evalu ed for increas ng level of damage aft r the r moval of the load, starting from the load level at the first crack nucleation to the load level of incipient collaps . In rder to v rify the accuracy and reliability of the proposed compu ational model, th numerical r sults, obtained in t ms of variations of the vibration l characteristics of the system will lso b compared with xperimental ata r ported in literature. R sults show the pplicability and reliability of damag identification procedu es based on both mathematical models and experimental data for structural systems such s reinforced concrete beams common i girder bridges. © 2022 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review u der re ponsibility of MedFract2Guest Editors. K ywords: Str ctural Health Monitoring; Reinforced Concr te Beams; Simulation-Based Damage Detection; Modal Properties; Diffuse Cracking; Cohesive Fracture. 2nd Mediterranean Conference on Fracture and Structural Integrity A Cohesive fracture approach for the nonlinear analysis of load induced degradation of vibration characteristics in RC beams Andrea Pranno a , Fabrizio Greco a, *, Paolo Lonetti a , Daniele Gaetano a , Claudio Le Piane b , Umberto De Maio a a Department of Civil Engineering, University of Calabria, Via P. Bucci Cubo 39B, Rende 87036, Italy b Provincial Administration of Cosenza, Corso Telesio 17, 87100 Cosenza, Italy

* Corresponding author. Tel.: +390984496916. E-mail address: fabrizio.greco@unical.it * Corresponding author. Tel.: +390984496916. E-mail address: fabrizio.greco@unical.it

2452-3216 © 2022 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0 ) Peer-review under responsibility of the MedFract2Guest Editors. 2452-3216 © 2022 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0 ) Peer-review u der responsibility of t MedFract2Guest Editors.

2452-3216 © 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the MedFract2Guest Editors. 10.1016/j.prostr.2022.05.070

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Nomenclature d

scalar damage function

E

Young’s modulus of the bulk elements

0 0 , n s K K normal and tangential interfacial elastic stiffness parameters , n s t t normal and tangential components of the cohesive traction vector m  mixed-mode displacement jump , n s   normal and tangential components of the displacement jump  Poisson’s ratio of the bulk elements    displacement jump between the crack faces p n K normal interfacial plastic stiffness parameter c n K normal interfacial plastic stiffness parameter (in compression) 

scalar factor which takes into account the partial contact between the crack surfaces

1. Introduction The study on how the damage phenomena affect the static and dynamic mechanical response is fundamental to reduce the risk of catastrophic failures of structures (Greco et al., 2013; Skrame et al., 2022). It has been demonstrated, through the recent progress in the Structural Health Monitoring field (SHM), that damage phenomena can seriously affect the mechanical properties of the structures, specifically in complex engineering applications (such as bridges) where the structural integrity should be monitored in real-time to guarantee adequate level of safety (Bruno et al., 2016; Lonetti and Pascuzzo, 2014). For this reason, the development of mathematical models of structural systems is of critical importance for identifying effectively the onset and evolution of damage phenomena (Lonetti et al., 2019). It is widely known that by monitoring and analyzing variations in the static and dynamic mechanical behavior of structures that are induced by damage phenomena it is possible to identify structural damage. In general, natural vibration frequencies, natural vibration modes, modal damping factors, and other structural parameters are often considered as indicators of integrity of the structures based on their modal characteristics (Doebling et al., 1998; Salawu, 1997). For instance, the modal parameters are global quantities which can be measured with the experimental modal analysis “EMA” (Penny, 1988; Piersol et al., 2010; Silva and Maia, 1999) as a result of an elaboration process of the structural dynamic response when the structure is subjected to artificial excitations and the operational modal analysis “OMA” (Brincker and Ventura, 2015; Rainieri and Fabbrocino, 2014) when it is subjected to environmental excitations. As "model-based" damage identification techniques we refer to a variety of techniques for detecting damage by comparing the dynamic response of a structure, predicted by a mathematical model able to detect the damage occurrence, with its vibration response observed experimentally on the monitored structure. In this context it is clear the importance of adopting affective methodologies to simulate the occurrence of damage phenomena. Similar to the case of homogeneous solids, damage in plain concrete structural elements is mainly characterized by the presence of single cracks and notch defects (Lin, 2004; Shen and Pierre, 1994; Sinha et al., 2002). On the other hand, simulating damage phenomena in reinforced concrete structural elements is more complicated since their nonlinear behavior, arising from its heterogeneity (Luciano and Willis, 2003) as well as the different damage scenarios, makes them more challenging to investigate. To take into account concrete nonlinearities due to cracking effects and interactions between the constituent materials, such as adhesion between the concrete and reinforcement steel bars (De Maio et al., 2020a), advanced mathematical theories are commonly employed for instance those based on continuum and discrete fracture approaches (Luciano and Sacco, 1998; Greco et al., 2022; Pascuzzo et al., 2020; Greco et al., 2021a). Additional brittle failures can be caused in heterogeneous materials by onset of instabilities (De Maio et al., 2020; Greco et al., 2020b), commonly observed due to reinforcing fibers acting at the nano-, micro- or macro scales (Raffaele Barretta et al., 2018; Greco et al., 2020a, 2018a, 2018b) and due to the adoption of kind of reinforcement with advanced microstructures such as bioinspired, functional and metamaterials (Ammendolea et al., 2021; Greco et al., 2021b, 2020c; Pranno et al., 2022). Because of its current and potential uses in many types of reinforcements, theoretical examination of free vibration behaviors of micro-beams is also a developing research topic

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(Acierno et al., 2017; Barretta et al., 2020, 2015; R. Barretta et al., 2018; Luciano, 2001). Additional damage phenomena associated with concrete materials can be caused by corrosion of the reinforcement steel, concrete cover separation and the debonding of the FRP system (De Maio et al., 2020b, 2019a). To examine the damage in reinforced concrete beams through parameters, such as damage location, damaged area, bending stiffness reduction, and damage evolution law, simplified models are available in the literature (Abdel Wahab et al., 1999). It is worth also noting that, the constraints conditions strongly effect the vibrational behavior of the reinforced concrete beams (simple (Cerri and Vestroni, 2000; Hanif et al., 2021; Koh et al., 2004a; Neild et al., 2002; Pešić et al., 2015; Voggu and Sasmal, 2021) or elastic (Casas and Aparicio, 1994) supports or free ends constraint (Van Den Abeele and De Visscher, 2000). As reported in the literature, several aspects of the mechanical behavior of reinforced concrete structures are highly dependent on the percentages of steel reinforcement, constraint conditions and geometry. For instance, in (Hamad et al., 2015) the numerical and experimental results performed on a reinforced concrete beam in a four-point bending test highlighted the natural vibration frequencies decrease as the level of damage increases reaching a maximum percentage reduction of about 16%. In (Casas and Aparicio, 1994) a simply supported reinforced concrete beams highlighted a natural vibration frequency reduction of about 25% and 15% for the first and the second vibration mode, respectively. Based on a numerical and experimental study by (Voggu and Sasmal, 2021), the first and fourth natural vibration frequencies were reduced by 55% and 29% due to the damage occurrence, respectively. Also in (Koh et al., 2004b) similar results have been obtained with a greater frequency reduction within 30% of the yielding load and with the fundamental vibration frequency reduced by 30% compared to the undamaged condition. The percentage decrease in natural vibration frequencies related to the maximum extent of the damage is about 40% in (Cerri and Vestroni, 2003) when the free ends constraint is introduced for the first mode shape, and about 33% and 24% for the second and fifth mode shapes respectively. In this study, using a cohesive fracture approach, a previous model presented by some of the authors (De Maio et al., 2020a) is extended to study the main factors influencing the static and dynamic response of reinforced concrete structures subjected to incremental static loading and unloading paths causing increasing levels of damage. Then, in order to investigate the degradation of vibration characteristics caused by the onset of cracking, a novel and advanced structural model is developed using the finite element method. Using interface elements, based on an advanced mixed-mode constitutive law, it is possible to account for both concrete plasticity, caused by cyclic tensile loading, and contact phenomena between crack surfaces. The contact phenomena induced by partial closure of the cracks caused by the presence of concrete aggregates were simulated using an adaptive formulation of the tangent stiffness at the point where the cohesive stresses change from tensile to compressive. A one-dimensional "truss" element is used to model and simulate the longitudinal and transverse steel reinforcement bars, and they are characterized by a purely axial mechanical behavior described by an elastoplastic constitutive law with linear hardening. One-dimensional "truss" elements are used to model the longitudinal and transverse steel reinforcement bars, and they are characterized by a purely axial elastoplastic mechanical behavior described by a constitutive law with linear hardening. Dynamical response is analyzed in terms of natural vibration frequencies and related modes shapes under quasi-static loading/unloading conditions of increasing amplitude, while the static mechanical response is analyzed in terms of load-displacement curves under monotonic and cyclic loading conditions. 2. Static and dynamic response of damaged structural systems under loading and unloading processes In the present work the degradation of vibration characteristics resulting from damage phenomena in reinforced concrete structures due to monotonic and cycling loading processes is investigated by using a numerical fracture model implemented in COMSOL Multiphysics® 5.6 finite element software. Two distinct mathematical models are incorporated: i) a diffuse interface model (DIM) that simulates the initiation and propagation of multiple cracks, and ii) an embedded truss model (ETM) to simulate the interaction between the concrete and steel reinforcements. The following works provide additional details on the study of the above-mentioned models (De Maio et al., 2020c, 2019b, 2022; Pascuzzo et al., 2022), whereas Section 2.1 describes the tensile concrete nonlinear cohesive law that was adopted to study the behavior of tensile concrete subjected to damage, plasticity, and contact phenomena. Additional details on the general formulation of the above-mentioned models can be found in the following works (De Maio et al., 2020c, 2019b, 2022, 2021), while the nonlinear cohesive law, adopted to simulate the mechanical behavior of concrete under tension subjected to damage, plasticity, and contact phenomena, is reported in the following section.

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2.1. Nonlinear traction-separation law for loading and unloading processes Concrete exhibits a linear elastic behavior in the pre-peak phase and a more complex softening behavior in the post-peak phase. According to well-known constitutive tensile found in the literature, tensile monotonic stresses have different constitutive relations, for example, a linear, bilinear and exponential softening are proposed in (Hillerborg et al., 1976), (Petersson, 1981) and (Sima et al., 2008), respectively. Other study were conducted for cycling loads in (Reinhardt et al., 1986) and (Foster and Marti, 2003). The proposed traction-separation law allows to consider the effects of damage in mixed-mode conditions, contact phenomena between crack surfaces and residual plastic deformations associated with the concrete damage behavior. The fracture initiation involves a quadratic stress criterion, the propagation is governed by a mixed-mode equation, while the damage is represented by an exponential damage parameter, which is determined by a function of the equivalent displacement, which is appropriate given the mixed-mode conditions. Furthermore, due to the partial closure of cracks, which is induced by the presence of concrete aggregates, a formulation which adapts the tangent stiffness is developed to take into account the inversion of the cohesive stresses from tensile to compression. The mixed-mode traction separation las is defined by the following expression:

   

    

max

n 





p

  

n       s t t

  

 

0

0

K

 1  



n

n

max    p

d

n

,

(1)

n

n

0

0

K

 

s

s 

max n n    is the plastic normal displacement, p

where s and n define tangential and normal directions, respectively,

max n  is the maximum normal displacement jump detected along the entire loading process and the scalar d is the damage variable. In terms of the latter, a linear-exponential evolution law can be defined as a function of the equivalent mixed-mode displacement jump:

2

2





  

,

(2)

m

n

s

n  yield the equivalent displacement independent from the compressive normal

where the Macaulay bracket

displacement jump:

max    0

         

0

m

m

      

      

  

  

max          0 f m m m 

  

0

1 exp

 

m

0



0      max m m m f

1   

1

d



m

,

(3)

  

max

1 exp



 

m

max   

f

1

m

m

with 0 m  denoting the equivalent displacement jumps at the onset of the damage and f p n K characterizes the unloading phase as reported in Fig.1: plastic stiffness

m  at the total decohesion. A

0 max

max (1 )

n n      p d K

p

K



(4)

n

n

n

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0 n K denoting the initial stiffness under tensile and compression stresses.

with

Fig. 1. Proposed traction-separation law for loading and unloading tensile cycle written in terms of normal displacement jump.

The following relations define the proposed cohesive law when the normal cohesive stresses swap from tension to compression:   0 0 0 c p p p n n n n n n compression n p n n n n K t K                  , (5)

p c n n K K K   c

p   

,

(6)

0

n

0

n

n

 1  

   

0

p

K

K

n

n

c

K



,

(7)

n

1



with  representing a scalar factor which allow to reduce the initial compression stiffness 0 n K as the level of damage increases to simulate the partial contact between the cracks surfaces due to the presence of aggregates. A quadratic nominal stress criterion governs the crack initiation:

2

2

t

  

    

  

t

n

1



s

,

(8)









max

max

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where max  is the critical normal stress and max  is the critical tangential stresses at the cohesive interfaces. An equivalent displacement jump 0 m  was defined (valid only for 0 n   ) as in the following:

2

1





0     0 0 m n s

m

,

(9)

   2 0 s  



2

0   m n

where m s n     is the ratio between the tangential and normal displacement jump. 2.2. Dynamic indicators for damage detection in reinforced concrete beams Based on the literature, in structural systems the variation of the natural vibration frequencies between the undamaged 0 j f and damaged d j f configuration can be taken into account as damage indicator and it is called the inverse eigenvalue sensitivity method (Balageas et al., 2006).:

0

d

f

f



j

j

.

(10)

0

f

j

An alternative strategy to investigate the damage effects on the mechanical behavior of structural systems is based on the comparison between the natural vibration modes before and after the occurrence of damage. In this regard, it is possible to use the so-called “Modal Assurance Criterion” (MAC) introduced by Allemang and Brown in 1982 (Allemang and Brown, 1982). This criterion was originally introduced to verify the degree of similarity (also called correlation) between the experimental and analytical modes. The generic component of the MAC matrix referring to the comparison between the modes in the undamaged configuration and those in the damaged one is defined as:       2 ,0 , ,0 ,0 , , T i j d ij T T i i j d j d MAC        (11)

,0 i  denotes the component vector of the

th i  natural vibration mode in the undamaged configuration th j  natural vibration mode in the damaged configuration. The

where

, j d  denotes the component vector of the

and

obtained values of ij MAC ranges from 0 to 1, specifically a value equal to unity corresponds to a perfect correlation between the two considered modes, while a value equal to zero indicates that there is no similarity between the modes. Since the damage in structural elements induces a variation in the natural vibration modes, through the MAC values it is possible to obtain a scalar measure of the damage as a function of the correlation between the natural vibration modes before and after the damage occurrence. Another effective damage indicator is the “Curvature Damage Factor” (CDF) (Abdel Wahab and De Roeck, 1999) which represents an average of the modal curvature difference at the generic coordinate th i  evaluated through the following expression:

1

N 

,0 ''    ij

''

CDF



,

(12)

, ij d

i

N

1

j



where the modal curvature is defined by means of the central difference method:

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7

( 1) i h       2 2 j ij

( 1) i 

j

''

ij 



.

(13)

3. Investigation of the damage effects on the static and dynamic response of RC structural elements An investigation of the static and dynamic behavior of a reinforced concrete beam was performed comparing the numerical results with the experimental ones described by (Hamad et al., 2015). Based on the combination of DIM and ETM models, a numerical model is proposed and implemented using the finite element software COMSOL Multiphysics® 5.6. The proposed approach incorporates nonlinear springs to simulate the bond-slip mechanism between steel reinforcement bars and concrete and diffuse interface elements to simulate the nucleation of diffuse fractures in reinforced concrete structures. The geometry configuration, the material parameters and the cohesive law parameters were reported in Fig. 2, Tab.1 and Tab.2, respectively. The beam is restrained at both ends by supports consisting of highly stiffened plates used to spread the stress from the constraint reactions and by linear elastic springs simulating the elastomeric pads whose stiffness is equal 1.1x107 N/m.

Fig. 2. Geometry configuration and boundary conditions of the four-point bending test investigated (all dimensions are expressed in mm)

Tab. 1: Material mechanical properties.

Young’s Modulus [GPa]

Compressive strenght [MPa]

Traction strenght [MPa]

Poisson’s ratio

Yield strength [MPa]

Material

Concrete

40.3 210 210

0.20 0.30 0.30

-

36.5

2.10

Plain Steel Ø6 Ribbed Steel Ø10

393.6

- -

- -

490

Tab. 2: Parameters of the proposed cohesive law.

Ic G [N/m]

IIc G [N/m]

max  [Mpa]

max  [Mpa]

0 n K [N/m 3 ]

0 s K [N/m 3 ]







1.3162e14 1.3162e14

2.1

4.2

150 1500 5 250 0.5

As numerical result of the proposed nonlinear model, the load-deflection curve during the loading process (blue curve) exhibits a typical trilinear behavior, in which two slope variations are markedly observed, coinciding with the beginning of the main fractures in concrete (point A) and the yielding of the tension reinforcing longitudinal bars

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(point B). We observed a slight variation between the numerical and experimental results from 0 to 4 mm of deflection, where the numerical model highlighted a slightly stiffer structural response due to the common toughening effect induced by the DIM which is discussed in depth in the following scientific papers (De Maio et al., 2020c, 2019b, 2019a).

Exp. envelope Loading path Unloading paths

L10

10 15 20 25 30 35 40

L9

L8

L7

L6

L5

Load [kN]

L4

L3

L2

0 5

L2'-L6'

L8' L9' L10' L7'

0

2

4

6

8

10

12

14

Deflection [mm]

Fig. 3. Load versus mid-span deflection curves compared with the envelope of the experimental results by (Hamad et al., 2015) during the loading and unloading phase.

In addition, ten damage levels, corresponding to a percentage of the maximum load level which is equal to 42.1 kN, were identified and listed in Tab.3 based on the data reported in (Hamad et al., 2015). For each level of damage, an unloading process was performed and the obtained load-deflection curve is reported in Fig. 3 by mean of a gray dashed line with the exception of the first level of damage L1 corresponding to the undamaged configuration.

Tab. 3: Percentages of maximum load (“Max. load %”) associated with the investigated damage levels (“damage level”).

Damage level Max. load %

L1

L2

L3

L4

L5

L6

L7

L8

L9

L10

0 0

15.90

22.73

27.87

35.10

47.27

59.17

71.13

83.43

95.07

Load [kN]

6.69

9.58

11.6

14.7

19.9

24.9

29.9

35.1

40

Subsequently, a modal analysis was carried out by superimposing a linearized eigenvalue problem over the solution of the quasi-static analysis whose results were reported in Fig.3. The modal analysis, giving the natural vibration frequencies of the damaged beam, was superimposed on the static solution at the unloading phase (Points L1'-L10') to consider the effects of partially closed cracks caused by contact phenomena. As can be seen in Fig. 4, the obtained natural vibration frequencies, expected to be influenced by the presence of diffuse damage, were normalized with respect to the frequencies related to the undamaged configuration. The normalized frequencies were plotted as a function of the levels of damage and compared with the experimental envelope reported in (Hamad et al., 2015). For the first and fourth natural vibration modes, the numerical and experimental results are in perfect agreement. However, a slight difference was detected in high-order natural vibration modes (6 th and 7 th modes), because they are typically most affected by measurement and dispersion errors. The highest deviation is less than 5% and it has been detected at

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highest damage level for the 7 th modes, but considering the experimental uncertainties given by the high-order natural vibration mode, is still an acceptable deviation. The curve trends highlighted that the investigated natural vibration modes show a decreasing trend with three different slopes: the slope that occurs between the percentage failure load 0% and 15% is slight, the slope that occurs between 15% and 50% is considerable, and the slope that occurs between 50% and 95% remains similar to the first.

Fig. 4. Normalized natural vibration frequencies of the modes 1, 4, 6, and 7 compared with the envelopes of the experimental data reported in (Hamad et al., 2015).

In order to evaluate the correlation between the modal shapes in the undamaged configuration L1' with those in damaged ones L2'-L10', the MAC coefficients were evaluated and reported in in Tab. 4. The coefficients were determined by using formula (11) applied to mode shapes normalized with respect to the mass matrix and evaluating the displacements relating to mode 1, 4, 6, 7 and 8 along the top of the beam at 51 points equally spaced to each other 60 mm. It worth noting that, as the level of damage increases, the correlation between the mode in the undamaged configuration and that relating to the damaged configuration decreases, as a consequence of the fact that the values of the diagonal coefficients of the MAC matrix (MAC ii ), deviate more from the unit value as the damage level increases. These deviations are not particularly relevant, in fact the relative maximum percentage deviation measured is equal to 2.11% corresponding to the level of damage L10 ' for the 8 th mode. In Fig. 5, we then investigated the Curvature Damage Factor (CDF) normalized with respect to the CDF which is related to the highest damage level (L10'). Such indicator was determined by summing the absolute variations of the Modal Curvatures (evaluated in fifty points placed on the whole length of the top beam and equally spaced) between the undamaged configuration L1' and the other investigated damage levels L2' - L10' over all the investigated mode shapes. It will be noted that, at the undamaged configuration (L1'), the CDF is equal to zero, but it oscillates for the subsequent damage levels, indicating the presence of damage localization. The obtained curves show several peaks in the center of the steel-reinforced concrete beam whose intensity is related to the damage level and therefore to the fracture length. Furthermore, CDF indicator was shown to be heavily affected by the local effect caused by the loads and stiff plates. In fact, we can see noise in the oscillatory trend of the CDF in these beam zones.

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Tab. 4: MAC indicator evaluated for the different damage levels and with reference to the undamaged configuration.

L1’-L2’ 0.0000 0.0208 0.9998 0.0942 0.0347 L1’-L4’ 0.0000 0.0159 0.9972 0.0849 0.0448 L1’-L6’ 0.0000 0.0169 0.9965 0.0834 0.0484 L1’-L8’ 0.0000 0.0168 0.9969 0.0863 0.0460 L1’-L10’

L1’-L1’ 0.0000 0.0234 1.0000 0.0976 0.0320 L1’-L3’ 0.0000 0.0166 0.9981 0.0869 0.0418 L1’-L5’ 0.0000 0.0162 0.9966 0.0834 0.0476 L1’-L7’ 0.0000 0.0173 0.9969 0.0854 0.0468 L1’-L9’ 0.0000 0.0162 0.9965 0.0858 0.0466

1.0000 0.0128 0.0000 0.0010 0.0024 0.9989 0.0180 0.0000 0.0011 0.0025 0.9980 0.0199 0.0000 0.0012 0.0026 0.9975 0.0208 0.0000 0.0012 0.0027 0.9968 0.0221 0.0000 0.0012 0.0028

0.0128 1.0000 0.0234 0.0047 0.0264 0.0109 0.9973 0.0262 0.0018 0.0240 0.0096 0.9952 0.0242 0.0011 0.0227 0.0089 0.9956 0.0239 0.0014 0.0232 0.0083 0.9948 0.0253 0.0014 0.0230

0.0010 0.0047 0.0976 1.0000 0.4032 0.0006 0.0054 0.1235 0.9945 0.4001 0.0007 0.0034 0.1436 0.9877 0.3432 0.0008 0.0017 0.1471 0.9839 0.3264 0.0009 0.0018 0.1465 0.9825 0.3267

0.0024 0.0264 0.0320 0.4032 1.0000 0.0024 0.0341 0.0441 0.3524 0.9933 0.0020 0.0363 0.0315 0.3717 0.9919 0.0019 0.0357 0.0262 0.3690 0.9887 0.0019 0.0352 0.0242 0.3552 0.9829

0.9999 0.0145 0.0000 0.0011 0.0024 0.9984 0.0191 0.0000 0.0011 0.0025 0.9977 0.0204 0.0000 0.0012 0.0027 0.9972 0.0215 0.0000 0.0012 0.0027 0.9962 0.0231 0.0000 0.0012 0.0028

0.0123 0.9997 0.0254 0.0037 0.0261 0.0102 0.9959 0.0253 0.0013 0.0231 0.0091 0.9952 0.0235 0.0011 0.0228 0.0086 0.9953 0.0247 0.0014 0.0232 0.0077 0.9937 0.0258 0.0012 0.0228

0.0008 0.0052 0.1039 0.9990 0.4161 0.0007 0.0047 0.1339 0.9919 0.3740 0.0008 0.0021 0.1497 0.9836 0.3231 0.0008 0.0017 0.1455 0.9839 0.3298 0.0009 0.0021 0.1439 0.9805 0.3253

0.0026 0.0290 0.0398 0.3801 0.9985 0.0022 0.0357 0.0389 0.3577 0.9924 0.0019 0.0364 0.0270 0.3792 0.9906 0.0019 0.0356 0.0260 0.3594 0.9858 0.0019 0.0350 0.0225 0.3489 0.9789

0.0000 0.0161 0.9960 0.0855 0.0478

Fig. 5. CDF evaluated for the mode shapes 1, 4, 6, and 7 for all the investigated levels of damage during the unloading phase

4. Conclusions Static and dynamic loads applied monotonically and cyclically can damage reinforced concrete beams commonly used in girder bridges by inducing diffuse cracking effects. Typically, the damage will extend over a larger area as the load level increases until it affects the entire system. Therefore, it is essential to develop a damage model that can

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simulate the nonlinear phenomena of diffuse propagation of multiple cracks incorporating other nonlinearities of the materials constitutive laws to model the plasticity and heterogeneity of the materials and the adhesion between steel reinforcement and concrete. Then, in the present study, an advanced and innovative finite element model was developed to investigate how the diffuse cracking phenomenon affects vibration characteristics in reinforced concrete beams. Specifically, with the developed numerical formulation, the damage phenomena caused by quasi-static monotonic and cyclic loadings was determined and the evolution of the dynamic properties of reinforced concrete beams was investigated. The deduced vibration characteristics of the investigated beams highlighted that diffuse damage plays a large role on their dynamic behavior, even at low frequencies. Natural vibration frequency variation is strictly correlated, in addition to damage assessment, with modal deformations resulting from the most damaged system components. Natural vibration frequency is highly sensitive to damage evolution in reinforced concrete structures. Therefore, the frequency-based damage identification parameters can be used even when load and, as a consequence, the damage are moderate. Additionally, numerical results have established that the use of advance damage indicators, such as those based on modal curvature, allows for realistic detection of the extent and location of the damage. 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