# Issue 54

Frattura ed Integrità Strutturale, 54 (2020); International Journal of the Italian Group of Fracture

Table of Contents

T. I. J. Brito, D. M. Santos, F. A. S. Santos, R. N. da Cunha, D. L. N. F. Amorim https://youtu.be/Bgx5m9YrJbU On the lumped damage modelling of reinforced concrete beams and arches ………………….... 1-20 A. Boulebd, F. Noureddine, B. Mohcene, H. A. Mesbah https://youtu.be/lBC_Ovu7XlQ Modeling of CFRP strengthened RC beams using the SNSM technique, proposed as an alternative to NSM and EBR techniques ……………………………………………... 21-35 A. Kumar K, M. Reddy D https://youtu.be/mIsVoQ3cdkg A Novel Method to Estimate the Damage Severity Using Spatial Wavelets and Local Regularity Algorithm ……………………………………………………………… 36-55 E. M. Strungar, D. S. Lobanov https://youtu.be/FO36um9_pKI Mathematical data processing according to digital image correlation method for polymer composites 56-65 F. S. Brandão, L. F. F. Miguel https://youtu.be/Vj7kjfPQ-s4 Vibration control in buildings under seismic excitation using optimized tuned mass dampers ….. 66-87 R. B. P. Nonato https://youtu.be/K7EtKYFJJB4 Bi-level Hybrid Uncertainty Quantification in Fatigue Analysis: S-N Curve Approach ……... 88-103 O. Shallan, H. M. Maaly, M. M. Elgiar, A. A. El-Sisi https://youtu.be/gskW5eVOV8M Effect of stiffener characteristics on the seismic behavior and fracture tendency of steel shear walls 104-115 J. Akbari, A. Abed https://youtu.be/K529c0DgblQ Experimental evaluation of effects of steel and glass fibers on engineering properties of concrete … 116-127 I. El-Sagheer, M. Taimour, M. Mobtasem, Amr A. Abd-Elhady, H. El-Din M. Sallam https://youtu.be/JjwBauc6xrw Finite Element analysis of the behavior of bonded composite patches repair in aircraft structures 128-135

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Fracture and Structural Integrity, 54 (2020); ISSN 1971-9883

Z. Xiong, Y. Liu, B. Liu, L Jiang https://youtu.be/HE329iSOhJ8 Tension Strength Prediction of Transverse Branch Plate-to-Rectangular Joint with Concrete Filling …………………………………………………………………………… 136-152 I. Chekalil, A. Miloudi , M.P. Planche, A. Ghazi https://youtu.be/GgiGyc5_V7g Prediction of mechanical behavior of friction stir welded joints of AA3003 aluminum alloy …… 153-168 P. Jinlong, L. Guanhua, C. Jingming https://youtu.be/CQhJ_haiYO8 Numerical analysis of circular and square concrete filled aluminum tubes under axial compression 169-181 P. Livieri, F. Segala https://youtu.be/kuiCF9pKeMY A closed form for the Stress Intensity Factor of a small embedded square-like flaw…………… 182-191 V. Shlyannikov, A. Tumanov, R. Khamidullin https://youtu.be/4DXTpA0o1mk Strain-gradient effect on the crack tip dislocations density ………………………………… 192-201 M. Belaïd, M. Malika, S. Mokadem, S. Boualem https://youtu.be/TBXRmlm0r58 Probabilistic elastic-plastic fracture mechanics analysis of propagation of cracks in pipes under internal pressure ………………………………………………………………....… 202-210 M.A. Warda, H.S. Khalil, Seleem S. E. Ahmad, I.M. Mahdi https://youtu.be/QgnS4uCJfDc Optimum sustainable mix proportions of high strength concrete by using Taguchi method …... 211-225 A. Moslemi Petrudi, K. Vahedi, M. Rahmani, M. Moslemi Petrudi https://youtu.be/LlgwWMeKYBY Numerical and analytical simulation of ballistic projectile penetration due to high velocity impact on ceramic target …………………………………………………………………... 226-248 B Bartolucci, A. De Rosa, C. Bertolin, F. Berto, F. Penta, A.M. Siani https://youtu.be/IST-0tqTMDA Mechanical properties of the most common European woods: a literature review …………… 249-274 T. Nehari, K. Bahram, D. Nehari, D. Marni Sandid https://youtu.be/brcLnLXEkhc Numerical analysis of the influence of maximum residual thermal stresses on the intensity factor between the matrix and particle interfaces in metal matrix composite ………………………. 275-281 F. Benaoum, F. Khelil, A. Benhamena https://youtu.be/mvTp-gfeD2I Numerical analysis of reinforced concrete beams pre-cracked reinforced by composite materials …. 282-296

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Frattura ed Integrità Strutturale, 54 (2020); International Journal of the Italian Group of Fracture

A. Sirico, P. Bernardi, B. Belletti, A. Malcevschi, L. Restuccia, G. A. Ferro, D. Suarez-Riera https://youtu.be/Hq88STfV0sI Biochar-based cement pastes and mortars with enhanced mechanical properties ……………... 297-316 A. Golsoorat Pahlaviani, A. M. Rousta, P. Beiranvand https://youtu.be/U4dYA5PyVCo Investigation of the effect of impact load on concrete-filled steel tube columns under fire ………… 317-324

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Fracture and Structural Integrity, 54 (2020); ISSN 1971-9883

Editorial Team

Editor-in-Chief Francesco Iacoviello

(Università di Cassino e del Lazio Meridionale, Italy)

Co-Editor in Chief Filippo Berto

(Norwegian University of Science and Technology (NTNU), Trondheim, Norway)

Section Editors Marco Boniardi

(Politecnico di Milano, Italy)

Nicola Bonora Milos Djukic

(Università di Cassino e del Lazio Meridionale, Italy)

(University of Belgrade, Serbia)

Stavros Kourkoulis

(National Technical University of Athens, Greece) (University Politehnica Timisoara, Romania)

Liviu Marsavina Pedro Moreira

(INEGI, University of Porto, Portugal)

Guest Editor

SI: Structural Integrity and Safety: Experimental and Numerical Perspectives

José António Fonseca de Oliveira Correia

(University of Porto, Portugal.)

Advisory Editorial Board Harm Askes

(University of Sheffield, Italy) (Tel Aviv University, Israel) (Politecnico di Torino, Italy) (Università di Parma, Italy) (Politecnico di Torino, Italy) (Politecnico di Torino, Italy)

Leslie Banks-Sills Alberto Carpinteri Andrea Carpinteri Giuseppe Ferro

Donato Firrao

Emmanuel Gdoutos

(Democritus University of Thrace, Greece) (Chinese Academy of Sciences, China)

Youshi Hong M. Neil James Gary Marquis

(University of Plymouth, UK)

(Helsinki University of Technology, Finland)

(Ecole Nationale Supérieure d'Arts et Métiers | ENSAM · Institute of Mechanics and Mechanical Engineering (I2M) – Bordeaux, France)

Thierry Palin-Luc Robert O. Ritchie Ashok Saxena Darrell F. Socie Shouwen Yu Cetin Morris Sonsino

(University of California, USA)

(Galgotias University, Greater Noida, UP, India; University of Arkansas, USA)

(University of Illinois at Urbana-Champaign, USA)

(Tsinghua University, China) (Fraunhofer LBF, Germany) (Texas A&M University, USA) (University of Dublin, Ireland)

Ramesh Talreja David Taylor John Yates Shouwen Yu

(The Engineering Integrity Society; Sheffield Fracture Mechanics, UK)

(Tsinghua University, China)

Regional Editorial Board Nicola Bonora

(Università di Cassino e del Lazio Meridionale, Italy)

Raj Das

(RMIT University, Aerospace and Aviation department, Australia)

Dorota Koca ń da Stavros Kourkoulis

(Military University of Technology, Poland) (National Technical University of Athens, Greece)

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Frattura ed Integrità Strutturale, 54 (2020); International Journal of the Italian Group of Fracture

Carlo Mapelli Liviu Marsavina

(Politecnico di Milano, Italy)

(University of Timisoara, Romania) (Tecnun Universidad de Navarra, Spain)

Antonio Martin-Meizoso

Raghu Prakash

(Indian Institute of Technology/Madras in Chennai, India)

Luis Reis Elio Sacco

(Instituto Superior Técnico, Portugal) (Università di Napoli "Federico II", Italy)

Aleksandar Sedmak

(University of Belgrade, Serbia)

Dov Sherman Karel Sláme č ka

(Tel-Aviv University, Tel-Aviv, Israel)

(Brno University of Technology, Brno, Czech Republic) (Middle East Technical University (METU), Turkey) (Ternopil National Ivan Puluj Technical University, Ukraine)

Tuncay Yalcinkaya

Petro Yasniy

Editorial Board Jafar Albinmousa Nagamani Jaya Balila

(King Fahd University of Petroleum & Minerals, Saudi Arabia)

(Indian Institute of Technology Bombay, India) (Indian Institute of Technology Kanpur, India)

Sumit Basu

Stefano Beretta Filippo Berto K. N. Bharath

(Politecnico di Milano, Italy)

(Norwegian University of Science and Technology, Norway) (GM Institute of Technology, Dept. Of Mechanical Engg., India)

Elisabeth Bowman

(University of Sheffield)

Alfonso Fernández-Canteli

(University of Oviedo, Spain) (Università di Parma, Italy)

Luca Collini

Antonio Corbo Esposito

(Università di Cassino e del Lazio Meridionale, Italy)

Mauro Corrado

(Politecnico di Torino, Italy) (University of Porto, Portugal)

José António Correia

Dan Mihai Constantinescu

(University Politehnica of Bucharest, Romania)

Manuel de Freitas Abílio de Jesus Vittorio Di Cocco Andrei Dumitrescu Riccardo Fincato Milos Djukic

(EDAM MIT, Portugal)

(University of Porto, Portugal)

(Università di Cassino e del Lazio Meridionale, Italy)

(University of Belgrade, Serbia)

(Petroleum-Gas University of Ploiesti, Romania)

(Osaka University, Japan)

Eugenio Giner Ercan Gürses

(Universitat Politecnica de Valencia, Spain) (Middle East Technical University, Turkey)

Ali Javili

(Bilkent University, Turkey) (University of Piraeus, Greece)

Dimitris Karalekas Sergiy Kotrechko Grzegorz Lesiuk Paolo Lonetti Carmine Maletta

(G.V. Kurdyumov Institute for Metal Physics, N.A.S. of Ukraine, Ukraine)

(Wroclaw University of Science and Technology, Poland)

(Università della Calabria, Italy) (Università della Calabria, Italy)

Sonia Marfia

(Università di Cassino e del Lazio Meridionale, Italy)

Lucas Filipe Martins da Silva

(University of Porto, Portugal)

Tomasz Machniewicz

(AGH University of Science and Technology)

Hisao Matsunaga Milos Milosevic Pedro Moreira

(Kyushu University, Japan)

(Innovation centre of Faculty of Mechanical Engineering in Belgrade, Serbia)

(University of Porto, Portugal) (University of Bristol, UK)

Mahmoud Mostafavi Vasile Nastasescu

(Military Technical Academy, Bucharest; Technical Science Academy of Romania)

Stefano Natali Andrzej Neimitz

(Università di Roma “La Sapienza”, Italy) (Kielce University of Technology, Poland)

(Karpenko Physico-Mechanical Institute of the National Academy of Sciences of Ukraine, Ukraine)

Hryhoriy Nykyforchyn

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Fracture and Structural Integrity, 54 (2020); ISSN 1971-9883

Pavlos Nomikos

(National Technical University of Athens) (IMT Institute for Advanced Studies Lucca, Italy)

Marco Paggi Hiralal Patil Oleg Plekhov

(GIDC Degree Engineering College, Abrama-Navsari, Gujarat, India) (Russian Academy of Sciences, Ural Section, Moscow Russian Federation)

Alessandro Pirondi Maria Cristina Porcu Dimitris Karalekas Luciana Restuccia Giacomo Risitano

(Università di Parma, Italy) (Università di Cagliari, Italy) (University of Piraeus, Greece) (Politecnico di Torino, Italy) (Università di Messina, Italy) (Università di Padova, Italy) (Università di Brescia, Italy) (Università di Napoli "Federico II")

Mauro Ricotta Roberto Roberti

Elio Sacco

Hossam El-Din M. Sallam

(Jazan University, Kingdom of Saudi Arabia) (Università di Roma "Tor Vergata", Italy)

Pietro Salvini Mauro Sassu

(University of Cagliari, Italy) (Università di Parma, Italy)

Andrea Spagnoli Ilias Stavrakas

(University of West Attica, Greece) (Lublin University of Technology)

Marta S ł owik Cihan Teko ğ lu Dimos Triantis Sabrina Vantadori Natalya D. Vaysfel'd Charles V. White

(TOBB University of Economics and Technology, Ankara, Turkey

(University of West Attica, Greece)

(Università di Parma, Italy)

(Odessa National Mechnikov University, Ukraine)

(Kettering University, Michigan,USA)

Shun-Peng Zhu

(University of Electronic Science and Technology of China, China)

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Frattura ed Integrità Strutturale, 54 (2020); International Journal of the Italian Group of Fracture

Frattura ed Integrità Strutturale is an Open Access journal affiliated with ESIS

Sister Associations help the journal managing Australia: Australian Fracture Group – AFG

Czech Rep.: Asociace Strojních Inženýr ů (Association of Mechanical Engineers) Greece: Greek Society of Experimental Mechanics of Materials - GSEMM India: Indian Structural Integrity Society - InSIS Israel: Israel Structural Integrity Group - ISIG Italy: Associazione Italiana di Metallurgia - AIM Italy: Associazione Italiana di Meccanica Teorica ed Applicata - AIMETA Italy: Società Scientifica Italiana di Progettazione Meccanica e Costruzione di Macchine - AIAS Poland: Group of Fatigue and Fracture Mechanics of Materials and Structures Portugal: Portuguese Structural Integrity Society - APFIE Romania: Asociatia Romana de Mecanica Ruperii - ARMR Serbia: Structural Integrity and Life Society "Prof. Stojan Sedmak" - DIVK Spain: Grupo Espanol de Fractura - Sociedad Espanola de Integridad Estructural – GEF Turkey: Turkish Solid Mechanics Group Ukraine: Ukrainian Society on Fracture Mechanics of Materials (USFMM)

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Fracture and Structural Integrity, 54 (2020); ISSN 1971-9883

Journal description and aims Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is the official Journal of the Italian Group of Fracture. It is an open-access Journal published on-line every three months (January, April, July, October). Frattura ed Integrità Strutturale encompasses the broad topic of structural integrity, which is based on the mechanics of fatigue and fracture and is concerned with the reliability and effectiveness of structural components. The aim of the Journal is to promote works and researches on fracture phenomena, as well as the development of new materials and new standards for structural integrity assessment. The Journal is interdisciplinary and accepts contributions from engineers, metallurgists, materials scientists, physicists, chemists, and mathematicians. Contributions Frattura ed Integrità Strutturale is a medium for rapid dissemination of original analytical, numerical and experimental contributions on fracture mechanics and structural integrity. Research works which provide improved understanding of the fracture behaviour of conventional and innovative engineering material systems are welcome. Technical notes, letters and review papers may also be accepted depending on their quality. Special issues containing full-length papers presented during selected conferences or symposia are also solicited by the Editorial Board. Manuscript submission Manuscripts have to be written using a standard word file without any specific format and submitted via e-mail to gruppofrattura@gmail.com. Papers should be written in English. A confirmation of reception will be sent within 48 hours. The review and the on-line publication process will be concluded within three months from the date of submission. Peer review process Frattura ed Integrità Strutturale adopts a single blind reviewing procedure. The Editor in Chief receives the manuscript and, considering the paper’s main topics, the paper is remitted to a panel of referees involved in those research areas. They can be either external or members of the Editorial Board. Each paper is reviewed by two referees. After evaluation, the referees produce reports about the paper, by which the paper can be: a) accepted without modifications; the Editor in Chief forwards to the corresponding author the result of the reviewing process and the paper is directly submitted to the publishing procedure; b) accepted with minor modifications or corrections (a second review process of the modified paper is not mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. c) accepted with major modifications or corrections (a second review process of the modified paper is mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. d) rejected. The final decision concerning the papers publication belongs to the Editor in Chief and to the Associate Editors. The reviewing process is usually completed within three months. The paper is published in the first issue that is available after the end of the reviewing process.

Publisher Gruppo Italiano Frattura (IGF) http://www.gruppofrattura.it ISSN 1971-8993 Reg. Trib. di Cassino n. 729/07, 30/07/2007

Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0)

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Frattura ed Integrità Strutturale, 54 (2020); International Journal of the Italian Group of Fracture

NEWS from FIS

Dear friends, we are pleased to announce that we activated two new services for our Journal: - It is possible to type your comments to the published papers, opening a new “social” era for our journal. Obviously, in order to write a comment, it will be necessary to login using your credentials. Below the Abstract you find the field to be filled.

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Finally, we introduced a supplementary step in the reviewing process. After the “normal” blind peer review, the papers are pre-published in the “Online first” section of the journal. Well, all the readers are solicited to send their comments and, if

necessary, authors will be requested to modify the paper before the final publication. Please, do not hesitate to send us your suggestions to further improve our journal. Very best,

Francesco Iacoviello Frattura ed Integrità Strutturale Editor in Chief

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T. I. J. Brito et alii, Frattura ed Integrità Strutturale, 54 (2020) 1-20; DOI: 10.3221/IGF-ESIS.54.01

On the lumped damage modelling of reinforced concrete beams and arches

Thalyson Issac de Jesus Brito, Danilo Menezes Santos, Fabio Augusto Silva Santos, Rafael Nunes da Cunha, David Leonardo Nascimento de Figueiredo Amorim Laboratory of Mathematical Modelling in Civil Engineering, Post-graduation Programme in Civil Engineering, Department of Civil

Engineering, Federal University of Sergipe, São Cristóvão, Brazil britothalyson@gmail.com, http://orcid.org/0000-0003-0884-2950 dmsantosse@gmail.com, http://orcid.org/0000-0002-3165-7553 fabioaugustoufs@gmail.com, http://orcid.org/0000-0003-0554-2119 fael3005ufs@academico.ufs.br, http://orcid.org/0000-0003-2503-6758 david.amorim@ufs.br, http://orcid.org/0000-0002-9233-3114

A BSTRACT . The analysis of reinforced concrete structures can be performed by means of experiments or numerical studies. The first way is usually quite expensive, so the second one sometimes is a good option to understand the physical behaviour of actual structures. Lumped damage mechanics appears as one of the latest nonlinear theories and presents itself as an interesting alternative to analyse the mechanical behaviour of reinforced concrete structures. The lumped damage mechanic applies concepts of the classic fracture and damage mechanics in plastic hinges for nonlinear analysis of reinforced concrete structures. Therefore, this paper deals with a novel physical definition of the correction factor γ for cracking evolution that ensures the presented lumped damage model depicts accuracy when it is compared to experimental observations of reinforced concrete beams and arches. Based on such experiments, the numerical analysis showed that γ value has upper and lower thresholds, depending on the physical and geometric properties of the reinforced concrete element. Notwithstanding, for γ values inside of the proposed interval, there is a best value of γ . K EYWORDS . Plastic hinges; Lumped damage mechanic; Reinforced concrete; Beams; Arches.

Citation: Brito, T.I.J., Santos, D.M., Santos, F.A.S., Cunha, R.N., Amorim, D.L.N.F., On the lumped damage modelling of reinforced concrete beams and arches, Frattura ed Integrità Strutturale, 54 (2020) 1-20.

Received: 13.04.2020 Accepted: 29.06.2020 Published: 01.10.2020

Copyright: © 2020 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

I NTRODUCTION

einforced concrete structures are worldwide used. In order to understand the behaviour of reinforced concrete structures due to different loading conditions, several experimental researches were carried out during the last decades. For instance, Vecchio and Emara [1] analysed a reinforced concrete frame with monotonic and cyclic loads. Carpinteti et al. [2] studied the scale effect in concrete specimens under uniaxial compression. Marsavina et al. [3] measured the chloride penetration in concrete structures. Sharifi [4] studied the influence of self-consolidating concrete in R

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T. I. J. Brito et alii, Frattura ed Integrità Strutturale, 54 (2020) 1-20; DOI: 10.3221/IGF-ESIS.54.01

reinforced concrete beams. Caratelli et al. [5] and Abbas et al. [6] analysed the performance of precast tunnel segments actually built in Italy [5] and Canada [6]. Finally, Ruggiero et al. [7] carried out full scale explosion experiments in reinforced concrete slabs. Since experimental research is usually quite expensive, numerical studies are often feasible to understand the physical behaviour of actual structures. Among several possibilities, theory of plasticity, fracture and damage mechanics are usually chosen. Theory of plasticity is probably the most known nonlinear principle. In such theory the inelastic effects are quantified by plastic strains e.g. the nonlinear behaviour of the reinforced concrete slabs during the direct contact explosion were modelled by Ruggiero et al. [7] throughout the theory of plasticity coupled with a three-dimensional finite element analysis. Fracture mechanics [8] were developed in order to quantify the propagation of discrete cracks in continuum media. For instance, Shi et al. [9] analysed a plain concrete tunnel lining with tens of thousands of finite elements with a fracture model. Note that the numerical response [9] is quite close to the experimental one [10]. Damage mechanics [11] introduces a variable, called damage, which quantifies the micro-crack density for concrete-like materials or micro-voids for metallic ones. In such theory, these micro-cracks are not small enough to be neglected but not big enough to be considered as discrete cracks. Several damage models were proposed in literature [12-20]. Despite the accuracy of the plastic, damage and fracture models, due to the material complexity of reinforced concrete structures, such models are usually not suitable for practical applications. Alternatively, lumped damage mechanics (LDM) applies some key concepts of classic fracture and damage mechanics in plastic hinges. LDM was firstly formulated for reinforced concrete structures under seismic loads [21-23]. Afterwards, LDM was rapidly expanded for other reinforced concrete structures [24-32], as well as steel frames [33-36], plain concrete tunnel linings [37] and masonry arches [37-38]. Note that LDM models present objective solutions [39-41]. For reinforced concrete structures the damage variable quantifies the concrete cracking and the plastic rotation variable accounts for the reinforcement yielding. Such damage variable takes values between zero and one. Therefore, the generalised Griffith criterion is used as an energy balance to crack propagation i.e. damage evolution. The plastic rotation evolution is accounted for a kinematic plastic law. Note that the evolution laws for damage and plastic rotation on the referred papers [21-32] and the references therein varies due to the purpose of each study and the applications per se . Perdomo et al. [23] proposed evolution laws that the model parameters can be easily associated to classic reinforced concrete theory. Despite such model’s [23] accuracy for bearing load capacity of reinforced concrete structures, service loads are often not well estimated. As an attempt to present a simple calibration of this model, Alva and El Debs [27] proposed a correction factor that enhances the damage evolution law leading to solutions that are even more accurate. However, Alva and El Debs [27] do not present the physical meaning of such factor, which might be an issue to this model reach practical engineering applications. In the light of the foregoing, this paper aims to propose a novel understanding of the physical meaning of the correction factor presented by Alva and El Debs [27]. Therefore, such correction factor is widely analysed using some experiments in order to deepen knowledge on the model and facilitate its application in practical engineering problems. Statics of circular arch elements onsider the structure depicted in Fig. 1a, which is composed by m circular arch elements connected by n nodes. External forces can be applied at each node, as illustrated for node j ( P uj , P wj and P θ j ). Note that the first index of the applied force describes its direction and the second one the node e.g. P wj is the force at the node j parallel to the Z - axis. Generically, the applied loads are gathered in the matrix of external forces, given by: 1 1 1 ... ... ... T u w ui wi i uj wj j un wn n P P P P P P P P P P P P P (1) where the superscript T means “transpose of”. Consider a circular arch element b between nodes i and j , as the one presented in Fig. 1. A circular arch element is defined by its arc angle ( χ b ), a radius ( R b ) and an angle between the Z-axis and z-axis ( β b ). Such element presents internal forces with respect to the global XZ system as depicted in Fig. 1b. Then, the matrix of internal forces is given as follows: T ui wi i uj wj j b Q Q Q Q Q Q Q (2) C L UMPED DAMAGE MECHANICS

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T. I. J. Brito et alii, Frattura ed Integrità Strutturale, 54 (2020) 1-20; DOI: 10.3221/IGF-ESIS.54.01

where Q ui is the internal force at the node i in the X -axis direction, Q wi is the internal force at the node i in the Z -axis direction and so on. Now, adopting the same notation proposed by Powell [42], another set of static variables is introduced (Fig. 1c) which are called as generalised stresses. Then, the matrix of generalised stresses gathers two bending moments at the edges of the element ( M i and M j ) and an axial force at the node i ( N i ) i.e. T i j i b M M N M (3)

(a)

(b) (c) Figure 1: (a) Structure composed by circular arch finite elements, (b) internal forces and (c) generalised stresses.

Regarding the equilibrium of the element, the internal forces and the generalised stresses can be expressed as [31]:

T b b b Q B M

(4)

where [ B ] b is the transformation matrix, given by:

3

b

b

sin

cos

1

b

b

R

R

sin

sin

b

b

b

b

sin

cos

B

0

b

b

b

R

R

sin

sin

b

b

cos 1 sin sin b b

cos 1 cos sin b b

b

b

cos

sin

0

b

b

(5)

sin

cos

b

b

b

b

b R b R

b

b

b

cos

cos

sin

cos

b

b

0

b

b

R

R

R b R

sin

sin

b

b

b

b

b

cos

b

b

b

b

b

b

b

sin

cos

cos

cos

sin

1

b

b

R

R

sin

sin

b

b

b

b

cos 1 sin b

cos 1 cos b

b

b

b

b

b

b

b

b

cos

sin

0

b

b

sin

sin

Then, the equilibrium relation of the structure is given by:

m

1 b P Q

(6)

E b

being { Q E } b the expanded internal forces matrix, given as follows:

E Q

0 0 0} T

1 {0 0 0 node ui wi i Q Q Q node i

uj Q Q Q wj j

(7)

b

node n

node j

Therefore, the equilibrium relation can be expressed as:

1 m T E b b b P B M

(8)

where [ B E ] b is the expanded transformation matrix, given in a similar way of { Q E } b . Kinematics of circular arch elements

Consider again the structure depicted in Fig. 1a, the node i presents two translations parallel to the global X - and Z -axes ( u i and w i , respectively) and a rotation in the XZ plane ( θ i ). Such translations and rotation are here called generalised displacements. Then, the generalised displacements of all nodes of the structure are gathered in the generalised displacements matrix, given by: 1 1 1 ... ... ... T i i i j j j n n n u w u w u w u w U (9) Now, the generalised deformations are defined as conjugate quantities of the generalised stress i.e. i , j and δ i are conjugated to M i , M j and N i , respectively. Such quantities are gathered in the generalised deformations matrix: T i j i b Φ (10)

Therefore, the mechanical power of the element is expressed as:

4

T T E Φ

(11)

M U Q M

b

b

b

b

Assuming small displacements and small deformations, the matrix of generalised deformations is given by: E b b Φ B U (12) Elasticity of circular arch elements Consider an elastic circular arch element with its internal forces at edges in the local coordinate system as the one depicted in Fig. 2a. Thus, the strain energy of the element is expressed by: 2 2 b M N where N ( ψ ) and M ( ψ ) are the normal force and bending moment for a cross section ψ , respectively, AE is the axial stiffness and EI is the flexural stiffness of the element. Note that N ( ψ ) and M ( ψ ) can be obtained by equilibrium (Fig. 2b): ( ) ( 1 cos ) ( ) (1 cos ) sin sin sin ( cos ) ( ) cos sin i j i b b i i b b i i b i b b j i b b M M n R M M N R M N R N R M N N R (14) 0 2 2 b b R d U EI AE (13)

Figure 2: Elastic circular arch element: (a) internal forces and (b) stresses at any cross section.

where V ( ψ ) is the shear force for a cross section ψ . By applying Castigliano’s Theorem, the elastic generalised deformations are given by:

2

2

2

U

U

U

U

b

b

b

b

e i e j e i i U b j

2 M M M N M M

i

j

i

i

i

i M N j i b

2

2

2

U

U

U M

e Φ

0 F M

b

b

b

(15)

2

b

b

M M M

M N

M

b

i

j

j

i

j

U

2

2

2

b U N M N M N N b b i U U

b

2

i

i

j

i

i

5

being [ F 0 ] b the elastic flexibility matrix, expressed as follows [38]:

2

2

2

b U M M M N b U

U

b

2 M

i

j

i

i

i

2

2

2

U

U

U

0 F

b

b

b

2

b

M M

M N

M

i

j

j

i

j

2

2

2

b U M N M N N b U U

b

2

i

i

j

i

i

2

4 sin 3 2 cos

b

3sin cos

b

R U

2

sin cos

b

b

1 2

1 2

b

b

b

b

b

b

b

2

2

2

b

b

M

EI

R AE

sin

sin

i

b

b

b

sin cos

b

2 U M M b

R

2sin

sin cos

b

b

1 2

1 2

b

b

b

2

2

b

b

EI

R AE

sin

sin

i

j

b

2

2

5sin 5 cos

b

b

b

3 2 cos

b

R U

cos

2

1 2

b

b

b

b

b

b

2

M N

b

EI

sin

i

i

b

sin sin cos

b

b

b

cos

1 2

b

b

2

b

AE

sin

b b R U

sin cos

b

2

in cos

b

b

s

1 2

1 2

b

b j

b

2

2

2

b

b

M

EI

R AE

sin

sin

b

2

sin R U

b

b

b

sin cos

b

2

cos

1 2

b

b

b

b

2

M N

b

sin sin sin cos EI

j

i

b

b

b

b

cos

1 2

b

b

2

b

AE

sin

(16)

3 R U

2

b

b

b

b

b

2 3sin cos

b

3sin

cos

cos

2

b

b

b

b

2

2

b b

N

sin sin sin cos EI

i

b R

b

b

b

c

os

b

b

2

b

AE

sin

Deformation equivalence hypothesis For the sake of simplicity, consider initially a uniaxial problem where a damaged straight bar is axially loaded. By definition, the damage variable ω represents the micro-cracks density [11]. Then, such bar resists to the axial force by means of the effective stress i.e. the Cauchy stress σ divided by (1 – ω ). The following expression, named strain equivalence hypothesis [11], is given by substituting the effective stress in the Hooke’s law: 1 p E (17) where E is the Young’s modulus, ε is the total strain and ε p is the plastic strain. The previous relation can be rewritten as follows:

p

p e d

p

(18)

E E

E

1

1

6

Therefore, by assuming the strain equivalence hypothesis the total strain can be expressed by a sum of three parts: elastic, damaged and plastic ones. Note that if ω = 0 then ε d = 0; on the other hand, if ω tends to one, ε d tends to infinity, which represents that the straight bar is now broken into two parts. Analogously, for a circular arch element the deformation equivalence hypothesis can be defined as: e d p b b b b Φ Φ Φ Φ (19) being { Φ e } b , { Φ d } b and { Φ p } b the elastic, damaged and plastic generalised deformations matrices, respectively. For the lumped damage framework, { Φ d } b and { Φ p } b describe the inelastic effects that are concentrated at two hinges at the edges of the element (Fig. 3). In this paper it is assumed that the plastic deformations account for the reinforcement yielding and the lumped damage variables represent the concrete cracking. Assuming that the plastic elongations at both hinges can be neglected [38], the plastic generalised deformation matrix is given by: 0 T p p p i j b Φ (20)

where i p and j p are the plastic relative rotations at the edges i and j of the element, respectively. The damaged generalised deformation matrix is expressed as:

2 U d M

d

i

b

0

0

2

1

i

i

M

i

d

2

U M d d

d Φ

M

j

b

(21)

0

0

,

C

j

i

j

2

b

d M

1

i b N

b

b

j

j

0

0

0

being [ C ( d i , d j )] b the compliance matrix and d i and d j the lumped damage variables at the hinges i and j , respectively. Note that if there is no damage at the element then [ C ( d i , d j )] b is null; on the other hand, if the damage variables tend to one then the non-zero terms of [ C ( d i , d j )] b tend to infinity, which represents that the inelastic hinges are close to perfectly hinges. Then, by substituting (15), (20) and (21) in (19), the constitutive relation is given by: 0 , , p i j i j b b b b b b b d d d d Φ Φ F M C M F M (22)

where [ F ( d i , d j )] b is the flexibility matrix with damage, written as:

2

2

2

U U

U

1

b

b

b

2

M M M N

d

1

M

i

i

j

i

i

i

2

2

2

U U

U

1

b

b

b

b

, d d

(23)

F

i

j

2

1 M M d

M N

M

i

j

j

j

i

j

2

2

2

b U M N M N N b U U

b

2

i

i

j

i

i

7

Figure 3: Elastic circular arch element with two inelastic hinges.

Model reduction to straight elements A particular characteristic of the presented lumped damage model is that the circular arch element can degenerate to a straight element [37]. Therewith, if R b tends to infinity then χ b tends to zero, R b sin χ b tends to the length of the element L b and β b becomes its orientation. Then, the transformation and flexibility matrices are given as follows:

b

b

b

b

sin

cos

sin

cos

1

0

L

L

L L

b

b

b

b

b

b

b

b

sin

cos

sin

cos

B

lim

0

1

b

L

L

L L

R

b

b

b

b

b

b

sin 0 cos

b

sin 0

cos

b

b

(24)

L

L

1

b

b

0

1 3

d EI

EI

6

i

L

L

1

b

b

d d

lim , F

0

i

j

1 3

EI

d EI

6

R

b

b

j

b L AE

0

0

Damage evolution law The complementary energy of the element is given by: 1 1 , T T p i j W d d M Φ Φ M F

M

(25)

b

b

b

2

2

b

b

Since the lumped damage variable accounts for the concrete cracking, its evolution law is given by the generalised Griffith criterion for each hinge i.e.

8

d

G Y d

0

2

2

2 1 M U W G d M d M U W G d M 2 2 2 2 2 i b i i i j

i

i

i

i

G Y d

d

0

i

i

i

(26)

j

d

G Y d

0

j

j

b

j

j

2

G Y d

d

0

d

2 1

j

j

j

j

j

where G i and G j are the damage driving moments for the hinges i and j , respectively, and Y ( d i ) and Y ( d j ) are the cracking resistance functions. The cracking resistance function for a hinge i was obtained by means of experiments on reinforced concrete beams [23]:

ln 1 1

d

Y d Y q

i

(27)

i

0

d

i

being Y 0 and q model parameters. In order to generalise the previous equation, Alva and El Debs [27] proposed the following relation:

d

ln 1

Y d Y q

i

exp 1

d

(28)

i

i

0

d

1

i

where γ is an empiric parameter introduced as an attempt to ensure a better fitting between numerical and experimental responses for service loads. Plastic evolution law The plastic evolution laws for each hinge of the element are given by:

0 0 0 0 0 0 p i i p i i p j j p j f f f f j

M f

p

i

0 i C k

0

i

d

1

0

i

(29)

M

j

p

0 j C k

f

0

j

d

1

0

j

being f i and f j the yield functions for the hinges i and j , respectively, and C and k 0 model parameters.

M ODEL ASSOCIATION WITH REINFORCED CONCRETE THEORY

A

n evident advantage of the lumped damage models for reinforced concrete structures is that the model parameters can be easily associated to the classic reinforced concrete theory. An engineer in practice knows how to calculate four key quantities for any reinforced concrete element: first cracking moment ( M r ), ultimate moment ( M u ), plastic moment ( M p ) and ultimate plastic rotation ( u p ) [43]. Considering an inelastic hinge i , cracks start to nucleate when the first cracking moment is reached i.e.

2 2 M U 2 0 r b r i M 2

M M d

Y

(30)

i

0

i

Then, the parameter Y 0 is defined as the first cracking resistance.

9

Note that the analysed cross section (hinge i ) has a load bearing capacity, therefore the ultimate moment ( M u ) is associated to a certain cracking level, which is quantified by the ultimate damage ( d u ). Therewith, the generalised Griffith criterion is expressed as:

2

2

d

ln 1

M U

u

u

b

exp 1

Y q

M M d d

d

(31)

i

u

i

u

u

0

2 1

2

2 M d

d

1

u

i

u

where q and d u are the unknown variables. Since the load bearing point ( d u , M u ) is the local maximum point of the moment-damage curve, another equation may be defined as:

M

0 0 2 1 1 ln 1 u u Y d q d

1 ln 1 d

i

exp 1

d

d

0

(32)

u

u

u

d

i M M i d d

u

i

u

Then, with Eqns. (31) and (32) the variables q and d u can be calculated. Note that the parameter q is related to an additional cracking resistance due to the reinforcement. Analogously to the load bearing point, the plastic moment ( M p ) is associated to the plastic damage ( d p ). Thus, d p is calculated by the generalised Griffith criterion i.e.

d

2 p

ln 1

2 M U

p

b

Y q

exp 1

M M d d

d

(33)

i

p

i

p

p

0

2

2 M d

d

1

2 1

p

i

p

At this stage, the reinforcement starts yielding, therefore:

M

p

p

0 0 f k

M M

(34)

i

p

i

i

0

d

1

p

where k 0 is defined as the effective plastic moment. Finally, for the load bearing capacity the plastic evolution law is expressed by:

M

1

p

p

u

i

u

0 k

M M

f

C

0

(35)

i

p

i

p

u d

u

1

being C then defined.

E XAMPLES

Cantilever beam [43] lórez-López et al. [43] presents the test of a cantilever reinforced concrete beam subject to a lateral concentrated load as illustrated in Fig. 4. The beam properties are shown in Tab. 1 and the yield and rupture tensions of the steel bars are 412 MPa and 520 MPa, respectively. The numerical analyses carried out in this example vary the value of γ . The comparison among experimental and numerical responses is shown in Fig. 5. F

Beam

f c ' (MPa)

A s

A' s

h (cm)

b (cm)

1

26.0 20 Table 1: Geometric and material proprieties of the beam [43] 3 Ø 12.5mm 3 Ø 12.5mm 20

10

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