Engineering Design Applications III: Structures, Materials and Processes (Advanced Structured Materials, 124) 3030390616, 9783030390617

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Advanced Structured Materials

Andreas Öchsner Holm Altenbach   Editors

Engineering Design Applications III Structures, Materials and Processes

Advanced Structured Materials Volume 124

Series Editors Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of Applied Sciences, Esslingen, Germany Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Porto, Portugal Holm Altenbach, Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

Common engineering materials reach in many applications their limits and new developments are required to fulfil increasing demands on engineering materials. The performance of materials can be increased by combining different materials to achieve better properties than a single constituent or by shaping the material or constituents in a specific structure. The interaction between material and structure may arise on different length scales, such as micro-, meso- or macroscale, and offers possible applications in quite diverse fields. This book series addresses the fundamental relationship between materials and their structure on the overall properties (e.g. mechanical, thermal, chemical or magnetic etc) and applications. The topics of Advanced Structured Materials include but are not limited to • classical fibre-reinforced composites (e.g. glass, carbon or Aramid reinforced plastics) • metal matrix composites (MMCs) • micro porous composites • micro channel materials • multilayered materials • cellular materials (e.g., metallic or polymer foams, sponges, hollow sphere structures) • porous materials • truss structures • nanocomposite materials • biomaterials • nanoporous metals • concrete • coated materials • smart materials Advanced Structured Materials is indexed in Google Scholar and Scopus.

More information about this series at http://www.springer.com/series/8611

Andreas Öchsner Holm Altenbach •

Editors

Engineering Design Applications III Structures, Materials and Processes

123

Editors Andreas Öchsner Faculty of Mechanical Engineering Esslingen University of Applied Sciences Esslingen, Baden-Württemberg, Germany

Holm Altenbach Chair of Engineering Mechanics Faculty of Mechanical Engineering Institute of Mechanics, Otto von Guericke University Magdeburg Magdeburg, Sachsen-Anhalt, Germany

ISSN 1869-8433 ISSN 1869-8441 (electronic) Advanced Structured Materials ISBN 978-3-030-39061-7 ISBN 978-3-030-39062-4 (eBook) https://doi.org/10.1007/978-3-030-39062-4 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Different engineering disciplines such as mechanical, materials, computer and process engineering provide the foundation for the design and development of improved structures, materials and processes. The modern design cycle is characterized by an interaction of different disciplines and a strong shift to computer-based approaches where only a few experiments are performed for verification purposes. A major driver for this development is the increased demand for cost reduction, which is also connected to environmental demands. In the transportation industry (e.g. automotive or aerospace), this is connected with the demand for higher fuel efficiency, which is related to the operational costs and the lower harm for the environment. A possible way to fulfil such requirements is lighter structures and/or improved processes for energy conversion. Another emerging area is the interaction of classical engineering with the health and medical sector. This further volume in this series gives an update on recent developments in the mentioned areas of modern engineering design application. We would like to express our sincere appreciation to the representatives of Springer, who made this volume possible. Esslingen, Baden-Württemberg, Germany Magdeburg, Sachsen-Anhalt, Germany

Prof. Andreas Öchsner Prof. Holm Altenbach

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Contents

Geometrical Characterization of a Lumbar Spine . . . . . . . . . . . . . . . . . Daniel Villaseñor-Chávez, Luis Héctor Hernández-Gómez, Juan Alfonso Beltrán-Fernández, Juan Carlos Hermida-Ochoa and Juan Luis Cuevas-Andrade

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Antigravity Device for Intravertebral Rehabilitation . . . . . . . . . . . . . . . Itzia Calalpa-Torres, Christopher René Torres-SanMiguel, Guillermo Urriolagoitia-Sosa, Alejandro Cuautle-Estrada, Beatriz Romero-Angeles and Guillermo Manuel Urriolagoitia-Calderón

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Simplified Test Bench Used to Reproduce Child Facial Damage During a Frontal Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alejandro Cuautle-Estrada, Christopher René Torres-SanMiguel, Guillermo Urriolagoitia-Sosa, Luis Martínez-Sáez, Beatriz Romero-Ángeles and Guillermo Manuel Urriolagoitia-Manuel Scientific Visualization as a Tool for Signal Processing in EEG Interpretation: Car Driver Sleeping State Detection for Advanced Cars Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ivica Kuzmanić, Igor Vujović, Joško Šoda and Maja Rogić Vidaković Determination, Validation, and Dynamic Analysis of an Off-Road Chassis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mateus Coutinho de Moraes and Miguel Ângelo Menezes Experimental Investigation of the Dynamics of a Ropeway Passing Over a Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Siegfried Ladurner, Markus Wenin, Daniel Reiterer, Maria Letizia Bertotti and Giovanni Modanese

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Contents

Environmental Fatigue Analysis of the Feedwater Piping System of a BWR-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laura Guadalupe Carbajal-Figueroa, Salatiel Pérez-Montejo, Alejandra Armenta-Molina, Gilberto Soto-Mendoza, Luis Héctor Hernández-Gómez, Juan Alfonso Beltrán-Fernández and Pablo Ruiz-López Structural Vibrations in a Building of a Nuclear Power Plant Caused by an Underground Blasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alejandra Armenta-Molina, Abraham Villanueva-García, Gilberto Soto-Mendoza, Salatiel Pérez-Montejo, Pablo Ruiz-López, Juan Alfonso Beltrán-Fernández, Luis Héctor Hernández-Gómez and Guillermo M. Urriolagoitia-Calderón Analysis of an Aircraft Impact on a Dry Storage Cask of Spent Nuclear Fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Edgar Hernández-Palafox, Juan Cruz-Castro, Yunuén López-Grijalba, Luis Héctor Hernández-Gómez, Guillermo Manuel Urriolagoitia-Calderón and Laura Guadalupe Carbajal-Figueroa

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Review of Electromagnetic Compatibility on Digital Systems of Nuclear Power Plants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Israel Abraham Alarcón-Sánchez, Roberto Linares-y-Miranda, Luis Hector Hernández-Gómez, Yunuén López-Grijalba, Alejandra Armenta-Molina, Laura Guadalupe Carbajal-Figueroa and Luis Alberto Arenas-Magos Effect of Beam Rigidity on the Lateral Stiffness of a One-Storey Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Meziane Chalah, Farid Chalah, Salah Eddine Djellab and Djillali Benouar Fundamental Transverse Vibration Circular Frequency of a Cantilever Beam with an Intermediate Elastic Support . . . . . . . . . 127 Lila Chalah-Rezgui, Farid Chalah, Salah Eddine Djellab, Ammar Nechnech and Abderrahim Bali Axial Fundamental Vibration Frequency of a Tapered Rod with a Linear Cross-Sectional Area Variation . . . . . . . . . . . . . . . . . . . . 133 Farid Chalah, Lila Chalah-Rezgui, Salah Eddine Djellab and Abderrahim Bali Vibration of an SDOF Representing a Rigid Beam Supported by Two Unequal Columns with One Mounted on a Flexible Base . . . . . 153 Lila Chalah-Rezgui, Farid Chalah, Salah Eddine Djellab, Ammar Nechnech and Abderrahim Bali

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Vibration Analysis of a Uniform Beam Fixed at One End and Restrained Against Translation and Rotation at the Second One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Farid Chalah, Lila Chalah-Rezgui, Salah Eddine Djellab and Abderrahim Bali Fundamental Vibration Periods of Continuous Beams with Two Unequal Spans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Farid Chalah, Lila Chalah-Rezgui, Salah Eddine Djellab and Abderrahim Bali Transverse Displacements of Transversely Cracked Beams with a Linear Variation of Width Due to Axial Tensile Forces . . . . . . . 183 Matjaž Skrinar Mechanical Properties and Formability Evaluation of AA5182-Polypropylene Sandwich Panels for Big Data Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Kee Joo Kim and Tae-Kook Kim Effects of Process Parameters on the Machining Process in Die-Sinking EDM of Alloyed Tool Steel . . . . . . . . . . . . . . . . . . . . . . . 215 Mohamed M. Bahgat, Ahmed Y. Shash, Mahmoud Abd-Rabou and Iman S. El-Mahallawi Carbon Fiber Reinforced Polymer (CFRP) Composite Materials, Their Characteristic Properties, Industrial Application Areas and Their Machinability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Dervis Ozkan, Mustafa Sabri Gok and Abdullah Cahit Karaoglanli Numerical Investigation on the Influence of Doping on Tensile Properties of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Vahid Ahani and Andreas Öchsner Application of Mechanical Tests to Determine the Properties of Foam Rubber for Mill Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 González Uribe Iván, Urriolagoitia Sosa Guillermo, Romero Ángeles Beatriz, Torres San Miguel Christopher René, Gutiérrez Lonche Liliana, Hernández Ramírez Aldo, Cuautle Estrada Alejandro and Urriolagoitia Calderón Guillermo Manuel Comparison of the Stress-Strain Relationship of Right and Pseudo-developable Helicoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Vladimir Jean Paul

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ATR-FTIR Analysis of Melamine Resin, Phenol-Formaldehyde Resin and Acrylonitrile-Butadiene Rubber Blend Modified by High-Energy Electron Beam Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Ivan Kopal, Juliána Vršková, Marta Harničárová, Ján Valíček, Darina Ondrušová, Ján Krmela and Peter Hybler Introduction of Two Analytical Theories as Applied to Developable Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Marina Rynkovskaya A Study of the Effect of the Quinacridone Pigment Content and Storage Time on the Process of Crystallization of Pre-oriented Polypropylene/Quinacridone Fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Vladimíra Krmelová, Ivan Kopal, Mária Gavendová, Ivan Labaj, Jan Krmela, Marta Harničárová, Jan Valíček, Peter Hybler and Tomáš Zatroch Adjusting the Plastic Zone Development in Steel Plate Shear Walls—A Finite Element Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Fereshteh Hassani and Zia Javanbakht

Geometrical Characterization of a Lumbar Spine Daniel Villaseñor-Chávez, Luis Héctor Hernández-Gómez, Juan Alfonso Beltrán-Fernández, Juan Carlos Hermida-Ochoa and Juan Luis Cuevas-Andrade

Abstract The design and development of a system, which is used in the geometrical characterization of a healthy or pathological curvature of the human spine, are discussed. The main objective is to avoid the application of invasive techniques that involve the application of radiation, such as X-rays or computerized tomography. For this purpose, a compact design has been proposed. It was manufactured with 3Dprinted components and common electronic devices such as Arduino or NANO© . The anthropometric characteristics of Mexican individuals have been taken into consideration. This device can be used with young and adult patients. Preliminary results have been discussed. Keywords Ultrasonic sensor · Spine · Lordosis · Printed circuit

Abbreviations PCB Printer circuit board PWM Pulse-width modulation USB Universal Serial Bus

D. Villaseñor-Chávez · L. H. Hernández-Gómez (B) · J. A. Beltrán-Fernández · J. L. Cuevas-Andrade Instituto Politécnico Nacional, Sección de Estudios de Posgrado e Investigación, Escuela Superior de Ingeniería Mecánica y Eléctrica, Edificio 5, 2do piso. Unidad Profesional Adolfo López Mateos. Col. Lindavista, Alcaldía Gustavo A. Madero, 07738 Ciudad de México, México J. C. Hermida-Ochoa Centro de Investigación y Laboratorio Biomecánico, Carmen No. 18, Chimalistac San Ángel, 01070 Ciudad de México, México © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_1

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1 Introduction The spine plays a fundamental role in the human musculoskeletal system. Its mechanical design allows an adequate performance for many years. However, alterations occur, such as lordosis that causes pain. The costs associated with this pathology are high [1]. The most used method for the evaluation of curvature of the spine in standing position is the radiological that can cause harmful effects on the individual due to the excessive use of X-rays. Other alternatives are photographic analysis, goniometry, electro-goniometry and flexible ruler [2, 3]. One of the main applications of such methods is for the evaluation of the deviation of kyphosis and lordosis diseases (Fig. 1). Lordosis is a sagittal curvature of the anterior convex rachis. The column has four physiological curvatures, two of them are outwards in the thoracic spine (at the level of the ribs) and in the sacral spine [4]. On the other hand, the hyperlordosis (increased curvature) may be caused by an anterior rotation of the pelvis, the upper part of the sacrum that takes an anteroinferior inclination through the hips. It causes an abnormal increase in the lumbar curvature. Regarding the kyphosis, it is the most frequent deformation of the spine [5]. It is generated by a posterior convexity of one or several segments of the spine and some alterations of the vertebrae that adopt a typical wedge shape [6, 7].

Fig. 1 Schematic representation of the kyphosis and lordosis pathologies

Geometrical Characterization of a Lumbar Spine

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2 Statement of the Problem Currently, low back pain affects 84% of the population of the world in developed countries. This generates a disability associated with low back pain. There are 10 million of disabilities in the USA every year [8]. In Mexico, it was the seventh cause of work absenteeism in 2015 and it was placed second in this year. It is in accordance with the data of the Mexican Institute of Social Security, which is one of the main institutions of the National Health Service in our country. Thirteen percentage of the population goes to consultation for this condition. It can be multifactorial given the aetiology of pain [9]. In 2017, the Mexican Institute of Social Security reported 30,105 accidents associated with the vertebral column. Consequently, 2507 invalidation opinions were given for dorsopathies [10, 11]. In private hospitals in Mexico, 1827 patients were admitted due to dorsopathies during the period 2012–2014. The National Institute of Rehabilitation, which is a specialized hospital in these diseases in Mexico, reported 2,121 cases treated for dorsopathies in 2011. Their classification is presented in Table 1. Based on the aforementioned, an ultrasound sensing system has been proposed. It is related with the evaluation of lordosis and kyphosis diseases. Actually, these cases Table 1 Dorsopathies reported by the National Rehabilitation Institute 2014 [10] Dorsopathies

Frequency

Percentage

Global percentage

Dorsalgias

–

–

–

Lumbago

567

26.70%

2.61%

Radiculopathy

411

19.40%

1.89%

Lumbar disc disorders

164

7.70%

0.76%

Cervical

100

4.70%

0.46%

Cervical disc disorders

94

4.40%

0.43%

Lumbago with sciatica

77

3.60%

0.35%

Sciatica

44

2.10%

0.20%

Other dorsalgias

23

1.10%

0.11%

Deforming dorsopathies

–

–

–

Other deforming dorsopathies

307

14.47%

1.41%

Scoliosis

178

8.39%

0.82%

Other unclassified dorsopathies

8

0.38%

0.04%

Lordosis

2

0.09%

0.01%

Kyphosis

2

0.09%

0.01%

Spondylopathies

–

–

–

Spondylosis

124

5.85%

0.57%

Other spondylopathies

20

0.94%

0.09%

Total

2121

100%

9.77%

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are evaluated with invasive methods. However, they can affect the individuals in the long term. This is the case of radiography [12].

3 Materials and Methods The block diagram of the proposed system is illustrated in Fig. 2. Figure 3 shows the methodology, which was followed in the development of the sensing system, as well as the data acquisition system, the creation of the algorithm and the design of the printed circuit. The Nano V3 board was selected. This is a more specific product than the Arduino® one and Arduino® mega cards. It is compact and suitable for more synthetic applications. Despite its small size, it is ideal for the sensing system. It is made up of 14 digital input/output connectors, and six of them are used as PWM and 8

Fig. 2 Block diagram of the sensing system

Geometrical Characterization of a Lumbar Spine

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Fig. 3 Methodology followed in the development of the sensing system

analog input connectors. In addition, a mini-USB connection is used, instead of a classic USB port and a reset button (Fig. 4) [13]. In the following step, a comparison was made of the different distance sensors that are available. The specifications required by the system were taken into account. This is the case of compatibility, range, cost, size and power among others. Table 2 compares some of the proposed devices. The HC-SR04 ultrasonic sensor was selected as it meets the ideal characteristics for data acquisition [14–16]. The circuit design was performed in a computer program (Proteus© ), in accordance with the circuit shown in Fig. 5. Once the system was designed, the simulation

Fig. 4 Schematic arrangement of the Arduino Nano V3 card

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Table 2 Sensor comparison [13, 14] Model

Voltage (v)

Distance (cm)

GP2Y0A41SK0F

5

4–30

GP2Y0A21YK

3.1–0.04

10–80

GP2Y0A02YK

5

20–150

GP2Y0A710K0F

5

100–500

HY-SRF05

5

3–3000

HC-SR04

5

1–4000

LV-EZ0

5

40–600

IS471F

5

1–15

QRD1114

5

0–3

Fig. 5 Design of the sensing system

was carried out. The operation of the sensors in an ideal environment was observed, and its performance was evaluated in a virtual terminal. In the next step, the physical connections necessary for each component were implemented in a test plate. The physical functioning of the system was observed. The digital pins, which were required, were taken into account, since these will be used for the development of the algorithm. Figure 6 shows the connection of the components of the sensing system in a preliminary prototype. Communication tests between the Arduino Nano V3 card and the computer were made. The objective was to transfer the programming of the sensing system and visualize the data. For this purpose, the algorithm that allows communication between the

Geometrical Characterization of a Lumbar Spine

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Fig. 6 Sensing system on a plate test

hardware and the software was developed. Then, it was loaded into the programming card and performed the communication tests. The characteristics of the card of the sensing system are shown in Fig. 7.

Fig. 7 Arduino Nano V3 communication card

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4 Results Figure 8 shows the electronic design of the sensing system and its instrumentation. Simulation tests were performed, and the results were observed in the virtual machine (Proteus© ). Once the simulation tests were done, physical tests of the system were made on the test plate. Figure 9 illustrates the design of the printed circuit with its optimum characteristics for a correct operation of the sensing system. This will allow the reduction of the voltage drops caused by the thickness and length of the ideal tracks. In an initial test, the spine geometry of a male, who was 31 years old and apparently healthy, was evaluated (Fig. 10). Table 3 shows the coordinates evaluated. The geometry of the spine of the individual was obtained, for both lumbar lordosis and kyphosis. Figure 10 shows the coordinates of the points recorded with the sensing system. They are compared with the geometry of the spine. Good convergence has been observed.

5 Conclusions Engineering science allows the development of systems required in the area of biomechanics that are more complex and accurate for the pathological diagnosis of patients. As a result, a product with easy access and lower cost to society is offered. However, all criteria must be taken into account to obtain the desired result of these devices.

Fig. 8 Tests of the sensing system

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Fig. 9 Design of the printed circuit

Regarding the initial results of the design of the ultrasonic sensor, its disadvantages were evaluated. Such is the case of the multiple bounces of the emitter, environmental temperature and the black zone of the sensor that affects the measurement. The sensor signal in this area is not detected. The geometry of the spine was characterized with the proposed assembly. The results were summarized graphically. A convergence with real geometry was observed. These results are encouraging. Consequently, an optimization has been proposed in order to evaluate the deviation of the lordotic pathology.

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Fig. 10 Graphical representation of the spine of the individual

Table 3 Data recorded by the sensing system

X-coordinates (cm)

Y-coordinates (cm)

17.52

89.44

19.18

92.86

20.5

96.51

21.39

104.02

21.28

104.02

20.5

107.55

18.18

116.83

15.31

128.43

13.65

134.18

13.32

140.47

13.34

146.55

14.87

152.4

18.07

158.04

Geometrical Characterization of a Lumbar Spine

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Acknowledgements The authors kindly acknowledge the grant awarded by the National Polytechnic Institute and the National Council of Science and Technology.

References 1. Huynh, K.T. et al.: Development of a detailed human spine model with haptic interface. In: Haptics Rendering and Applications. IntechOpen. https://www.researchgate.net/publication/ 221923246_Development_of_a_Detailed_Human_Spine_Model_with_Haptic_Interface (2012) 2. Jackson, R.P. et al.: Lumbopelvic lordosis and pelvic balance on repeated standing lateral radiographs of adult volunteer and untreated patients with constant low Back pain. Spine (2000). https://doi.org/10.1097/00007632-200003010-00008 3. Willner, S.: Spinal pantograph-a non-invasive technique for describing kyphosis and lordosis in the thoraco-lumbar spine. Acta Orthopaedica Scandinavica (1981). https://doi.org/10.3109/ 17453678108992142 4. Espíndola, J.C.G., Viquez, A.F.P.: Cervical lordosis evaluation in asymptomatic volunteers from Navy Medical Center. Acta Ortopédica Mexicana. 22(1), 7–11 (2008) 5. Been, E., Kalichman, L.: Lumbar lordosis. The Spine Journal. 14(1), 87–97 (2014) 6. Fon, G.T. et al.: Thoracic kyphosis: range in normal subjects. American Journal of Roentgenology (1980). https://doi.org/10.2214/ajr.134.5.979 7. Herring, J.A.: Tachdjian’s Pediatric Orthopaedics E-book: From the Texas Scottish Rite Hospital for Children. Elsevier Health Sciences. Saunders, Philadelphia (2013) 8. Hartvigsen, J. et al.: What low back pain is and why we need to pay attention. The Lancet (2018). https://doi.org/10.1016/S0140-6736(18)30480-X 9. Aranda, M.F.S., et al.: Biomedical device for the prevention of low back problems through superficial electromiography. Pistas Educativas 40, 130 (2018) 10. Soto, M. et al.: Frequency of low back pain and its treatment at a Private Hospital in Mexico City. Acta Ortopédica Mexicana. 29(1), 40–45. http://www.ajronline.org/doi/10.2214/ajr.134. 5.979 (2015) 11. IMSS: Instituto Mexicano del Seguro Social. México: Informes Estadísticos. http://www.imss. gob.mx/conoce-al-imss/memoria-estadistica-2017 (2017). Accessed 29 June 2019 12. Ibarra, L.G.: Las enfermedades y traumatismos del sistema músculo esquelético. Un análisis del instituto nacional de rehabilitación de México, como base para su clasificación y prevención. Secretaría de Salud (2013) 13. McRoberts, M.: Beginning Arduino. Apress (2013) 14. Hughes, J.M.: Arduino a technical reference a handbook for technicians, engineers, and makers. O’Reilly Media, Inc (2016) 15. Zhmud, V.A. et al.: Application of ultrasonic sensor for measuring distances in robotics. Journal of Physics (2018). https://doi.org/10.1088/1742-6596/1015/3/032189 16. Freaks, E.: Ultrasonic ranging module hc-sr04. HC-SR04 datasheet. https://elecfreaks.com/ estore/download/EF03085-HC-SR04_Ultrasonic_Module_User_Guide.pdf (2016). Accessed 25 June 2019

Antigravity Device for Intravertebral Rehabilitation Itzia Calalpa-Torres, Christopher René Torres-SanMiguel, Guillermo Urriolagoitia-Sosa, Alejandro Cuautle-Estrada, Beatriz Romero-Angeles and Guillermo Manuel Urriolagoitia-Calderón

Abstract Low back pain is a serious health problem that affects a lot of people in productive age (López-Hernández in Análisis del Puesto de Trabajo de Policía para Investigar la Posible Etiología Laboral del Síndrome Doloroso Lumbar, Propuesta de Control. Universidad Instituto Politécnico Nacional, p. 13, 2010 [1]). In this research is presented an antigravity rehabilitation device, which is designed to allow the intervertebral decompression and the antigravity muscles rehabilitation of the lumbar area. In addition, the main components of the equipment such as the base, the frame and the iron where the patient lays down, is studied through a computational program (finite element method) that is used for developing a static-structural analysis. Finally, the automation of the equipment is shown since the execution times and movements used to decompression of the intervertebral discs and/or the rehabilitation of the spine (Asghar-Norasteh in Low Back Pain. InTech, pp. 33–34, 2012 [2]). Keywords Low back pain · Vertebrae · Antigravity · Rehabilitation · Mechanism

1 Introduction Low back pain is considered a public health problem in many parts of the world. This pain manifests itself between the lower limit of the ribs and the gluteal region (lumbar and sacral regions of the spine), whose intensity varies according to postures and physical activity of each person [2]. Most causes of low back pain are unknown; however, hypotheses have arisen that try to explain the mechanical alteration that generates pain. These include endurance of trunk extension, psychological stress, poor flexibility of the hip joint, poor muscle control of the trunk, inadequate posture and low body mass [3]. The main reason for this disorder is due to age, lifestyle, I. Calalpa-Torres · C. R. Torres-SanMiguel (B) · G. Urriolagoitia-Sosa · A. Cuautle-Estrada · B. Romero-Angeles · G. M. Urriolagoitia-Calderón Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica y Eléctrica, Sección de Estudios de Posgrado e Investigación, Unidad Profesional Adolfo López Mateos, Edif. 5, 2do piso. Col. Lindavista, 07320 Ciudad de México, Mexico e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_2

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situations and the type of work performed, however, mechanically the force of gravity is responsible for generating efforts in the spine, generating the problem of low back pain, therefore, the force exerted by gravity on the human body must be balanced by the continuous effort of the muscles on the skeleton, in order to oppose this force and keep the body in an upright posture [4], the muscles in charge of this activity and of counteracting the effects of gravity are the postural or anti-gravity muscles, however, the sedentary life in modern society produces the overuse of the postural muscles, favouring the development of muscular rigidity. This work proposes the design of an equipment that makes possible the intervertebral decompression and the rehabilitation of the anti-gravitational muscles of the lumbar area.

2 Methodology The main objective is to design an anti-gravitational column rehabilitation equipment to improve the quality of the patient, exercising the anti-gravitational muscles to help relieve the pain of the lumbar vertebrae. The steps used to perform the rehabilitation equipment of the anti-gravity type column are shown below. • Equip design This section shows the main characteristics of the equipment and its operation. • Numerical analysis of the mechanism The relevant numerical analysis of the equipment’s pieces where most of the loads are concentrated at the moment of the use of this is presented. • Engine and transmission The specific study for calculating the power and work of the engine is shown. • Automation of the rehabilitation device Automation of the work sequence through a Functional Graph of Stages and Transitions Control.

3 Development For the development of this rehabilitation mechanism, it is necessary to understand 2 concepts, the first concept is the antigravity that makes mention about the opposition to the gravity force of attraction that the earth exerts on the body [5]. The second concept includes the zero-gravity position or neutral body posture (NBP) that emerges from research conducted by NASA scientists who observed that the human body in zero gravity conditions takes this particular position (Fig. 1) with certain angles made by the joints which allowed the astronauts to be relieved of lower back pain and that the height of the intervertebral discs will increase. This information has allowed the development of new intervertebral disc decompression therapies that are used to

Antigravity Device for Intravertebral Rehabilitation

15

Fig. 1 Right, zero-gravity position, Left, position to decompress the vertebrae

treat various conditions such as disc bulging, herniation, degenerative disc disease, sciatica, spinal stenosis, spinal arthritis, among others in a safe manner [6–8]. The physical principle of zero-gravity position is planned to be used for rehabilitation therapy, however, for this principle to work, it is necessary for the human body to take a specific position in which, the bodyweight does not press the part of the lumbar and at the same time a separation of the lumbar vertebrae is generated. For this the Nachemson theory [9] is used, which shows the ideal position where the loads that comprise the weight of the human body have a deconcentration over the lumbar vertebrae, this position is subscribed as Supino, traction 500 N, which generates a null intradiscal pressure in an inclined position of 50° (Fig. 2). To corroborate this factor are considered data such as mass, normal and frictional force, which is defined as the tangential force acting between two bodies in contact that opposes or prevents movement, therefore, when two bodies they are at rest, there is a force of friction called static, for the particular case this force of friction is very useful because it prevents the body from sliding [10, 11]. In order to synthesize properly the rehabilitation of the lower back, it is necessary to separate the lumbar vertebrae and strengthen the lumbar muscles (psoas) that serve Fig. 2 Equipment in vertebra decompression position

Feet bra knee flexer

Transmision system

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as support for the lumbar area. Therefore, the equipment presented in Fig. 2 will support the patient, realising their anti-gravitational muscles. The process consists of positioning the patient at different angles of inclination on the X-axis, with the purpose of concentric isotonic contractions in the muscles of the back with the support of the proprioceptive system. The equipment has a transmission system which allows the bed to turn and be positioned at different angles, as required. It will also have a mechanism to flex the patient’s knees and allow him to take the zero-gravity position (Fig. 2).

3.1 Numerical Analysis In order to know if the structural materials are safety, the finite element method is used to carry out the static-structural study of the elements that are considered important for the safety and correct functioning of the equipment. Which are: • Base. • Plate. • Frameworks. For this analysis, two different materials are used, one for the axis and one for the plate and the frame, in Table 1 the mechanical properties of the materials are shown. Table 2 shows the results of the numerical analysis of the structural elements of the device. The numerical analysis shows stresses and elongations well below the elastic limit of the material, so it will no present problems the design, even it can be observed that values do not exceed yield limit. Table 1 Mechanical properties of materials

Materials

Steel AISI 4140

Steel AISI 1020

Model type

Isotropic linear elastic

Isotropic linear elastic

Elastic limit

4.150 × 108 N/m2

3.51571 × 108 N/m2

Young’s modulus

2.1 × 1011 N/m2

2 × 1011 N/m2

Poisson’s number

0.3

0.29

Density

7850 kg/m3

7900 kg/m3

Shear modulus

8×

7.7 × 1010 N/m2

Load applied to the base

1273.81 N

Load applied to the plate

1584.22 N

Load applied to the frame

2173 N

1010

N/m2

1273.81 N

Antigravity Device for Intravertebral Rehabilitation

17

Table 2 Numerical analysis results Component

Von Mises stress

Strain

Base

0

1.4346

2.5822

MPa

mm 0

0.003832

0.006898

Plate

MPa 0.0186

60.497

108.88

mm 0

11.571

28.828

Frame

MPa 1.38x10

-15

14.89

26.803

-5

58.846

mm 0

0.2279

0.4102

Axis

MPa 2.4533x10

32.692

mm 0

0.0061

0.0111

3.2 Engine Selection The requirements of the transmission system are calculated. The torque required to move the total weight of the person and equipment is assessed. Having a payload of 2452.5 N the torque required is 82.011 Nm. In this way, the power engine is obtained. Table 3 shows the results of the necessary engine power calculation. With these obtained values, the characteristic engine values are shown in Table 4.

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Table 3 Data and results to obtain the power

Table 4 Engine characteristics

Table 5 Gear motor specifications

Data

Results

T = F ×d F = 2452.5 N

P=

T = 2452.5 × 0.1038 = 254.56 Nm

P = 254.56 W

Time The time proposed to develop the work is 10 s

hp = 0.3412

254.56 Nm 1s

Engine Engine power

370 W

Engine speed

1725 rpm

Service factor

1

Nominal torque

2 Nm

Gear motor Engine power

120 W

Engine output speed

1695 rpm

Engine-reducer ratio

537.49:3.2

Output speed gear motor

3.2 rpm

Torque output gear motor

225 Nm

The selected engine has the required power according to the previously realized calculations, however, the present project requires working at lower revolutions and higher nominal torque, for this reason, a gearbox with a ratio of 395.46 is chosen to decrease revolutions and increase torque: Table 5 shows the Gear motor characteristics.

3.3 Band and Pulley Selection Once the engine data such as revolutions, power and torque were used to carry out the selection of the pulleys and the belt. First, information about the engine is gathered, such as the power in hp, the revolutions delivered by the engine unit and those required in the equipment (Table 6). • Step two is to determine the design power in hp for which it is required to know the service factor that is 0.8 and multiply it by the power in hp. Design power = 0.8 × 0.160 = 0.128 hp

Antigravity Device for Intravertebral Rehabilitation Table 6 Gear motor data

19

Power

0.160 hp

Gear motor revolutions

3.2 rpm

Required revolutions

3 rpm

• Selecting a belt that is within the range of 3V, 3VX. A speed relation is established, and it is obtained by dividing the speed of the gear motor between the required revolutions: 3.2 rpm = 1.06 3 rpm • Selecting of the driving pulley. The relationship between rpm and the diameter of the pulley determines the power that is needed to transmit the pulley. Therefore, the diameter will be 65 mm.

3.4 Programming of the Rehabilitation Sequence Next, the sequence of the rehabilitation process is described, which begin when the start button is activated, and the rehabilitation option is selected. This process takes place in three stages (Fig. 3): • First stage—When the start button is in state one. We proceed to introduce an angle of inclination and a time set giving the instruction that the engine is turned on.

BP 1.0

R 0.0

Start 0.0

0

1

0

Time lapse stablished by Angle the user 1.0 1.0 5 On engine B 3.0 8

BP 1.0 0

7

EQU

1

Equal Source A N7:10 Source N7:11

Angle sensor B 3.0 0

SPV B 3.0

On engine B 3.0

6

4

8

Input angle encoder numerical data converter MOV

1

Input angle encoder numerical data converter 0

Angle sensor B 3.0

1

6

Fig. 3 Rehabilitation sequence

Move Source Des1

HSC. N7:10

Angle sensor B 3.0 6

Time lapse stablished by the user TON Time On Delay Timer T4.4 Time Base 1.0 Pre-set 20

EN DN

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• Second stage—The patient reaches the set angle and the engine stops. When the timer complies with the time determined by the doctor, the engine is in the opposite direction. • Third stage—The engine stops when it reaches the vertical position (vertical position sensor). It is important to mention that, if you want to stop the treatment at some point of the routine, you can push the emergency stop button that detain the entire equipment.

4 Conclusions The anti-gravitational rehabilitation mechanism shows a simple operation, it considers the zero-gravity position allowing rest and relaxation of the lumbar muscles while strengthening them, making a comprehensive rehabilitation, giving support to the psoas muscles and separating the vertebrae to relieve pain. Thanks to the design considerations and its simple way of functioning, its operation is easy and can be adapted to any size of the human body, being applicable for children and adults. The materials proposed are appropriate for a useful life without problems of premature wear, seen in the numerical analyses made to structural elements, the yield limit is well above the factors obtained. While the engine and reducer selected are appropriate and their power is enough, which has a low cost for its development. Thanks to its slow movement but controlled by means of encoders, this mechanism is comfortable and easy to handle, making it possible to consider a future market study if it is desired to market to perform rehabilitation therapies of the spine. Acknowledgements The authors thanks for the support to National Polytechnic Institute (IPN) and National Council for Science and Technology (CONACYT) to make this work possible. The authors also thank the support of projects 1931 and 20196710, as well as an EDI grant, all by SIP/IPN.

References 1. López-Hernández, M.A.: Análisis del Puesto de Trabajo de Policía para Investigar la Posible Etiología Laboral del Síndrome Doloroso Lumbar, Propuesta de Control, Tesis de maestría, Universidad Instituto Politécnico Nacional, p. 13 (2010) 2. Asghar-Norasteh, A.: Low Back Pain (ed.), pp. 33–34. InTech (2012) 3. Mahecha-Toro, M.T.: Dolor lumbar agudo; Mecanismos, enfoque y tratamiento. Morfolia 3, 29 (2009) 4. Oliveira, C., Navarro-García, R., Ruiz-Caballero, J.A., Brito-Ojeda, E.: Biomecánica de la columna vertebral. Canarias Médica y Quirúrgica 4, 41–42 (2007) 5. Bosque, I. y Demonte, V.: Gramática Descriptiva de la Lengua Española, Volumen 3 (ed.) ESPASA, p. 5020 (2000) 6. Abhijeet, R. y Badge, S. T.: Inversion therapy & zero gravity concept; for all back-pain problems. J. Mech. Civil Eng. 18–22 (2014)

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7. Abhijeet, R.: Development and modelling of an apparatus for inversion therapy and zero gravity concept for all back-pain diseases. Int. J. Pure and Appl. Res. Eng. Technol. 2(9), 420–428 (2014) 8. Coleman, D., Rademakers, L., Reiny S., Schewerin, B. y Jones, J.: SPINOFF. NASA 60–61 (2013) 9. Nachemson, L.A.: Disc pressure measurements. Spine 6, 94–95 (1981) 10. Gowitzke, B.A. y Milner, M.: El Cuerpo y sus Movimientos; Bases Científicas (ed.), 90–95. Paidotribo (1999) 11. Nordin, M. y Frankel, V.H.: Biomecánica del Sistema Musculoesquelético (ed.), p. 282. McGraw-Hil (2004)

Simplified Test Bench Used to Reproduce Child Facial Damage During a Frontal Collision Alejandro Cuautle-Estrada, Christopher René Torres-SanMiguel, Guillermo Urriolagoitia-Sosa, Luis Martínez-Sáez, Beatriz Romero-Ángeles and Guillermo Manuel Urriolagoitia-Manuel

Abstract Around 186,300 children died from unintentional accidental injuries each year (Li et al. in Public Health 144:S57–S61, 2017 [1]). The main cause of death was head injuries. There is a lack of studies about face injuries in children due to a crash impact. The present research focuses on the design of an innovative test bench capable to assess damage in children during a vehicular collision trying to simulate a facial frontal impact. FMVSS 208 standard was considered to design the test bench. In addition, Asimov Morris methodology was used to build a simplified impact platform (Hollowell et al. in NHTSA Docket, 1999 [2]). Kinovea® software was applied to obtain speed and acceleration parameters for the test bench and these parameters by videogrametry. The outcome shows a simplified system that uses gravity force blending with elastic bands to reach a collision speed. These associations avoid the need to create a mechanism with large dimensions. Finally, analytical and experimental analysis was carried out to assess a specific facial injury in an artificial child head. Keywords Test bench · Frontal impact · Cranioencephalic injury · Passive safety · HIC · Videogrametry

1 Introduction The study of cranioencephalic injuries caused during a vehicular accident is very important because thanks to this it is possible to predict the possibilities of fatalities A. Cuautle-Estrada · C. R. Torres-SanMiguel (B) · G. Urriolagoitia-Sosa · B. Romero-Ángeles · G. M. Urriolagoitia-Manuel Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica y Eléctrica, Sección de Estudios de Posgrado e Investigación, Unidad Profesional Adolfo López Mateos, Edif. 5, 2do piso. Col. Lindavista, 07320 Ciudad de México, Mexico e-mail: [email protected] L. Martínez-Sáez Universidad Politécnica de Madrid, Instituto Universitario de Investigación del Automóvil, Campus Sur UPM, Carretera de Valencia (A-3), km7, 28031 Madrid, Spain © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_3

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that may occur, especially in children due to a lack of road responsibility culture. This work shows the development of a test bench capable to evaluating the head injury criteria Head Injury Criterion (HIC), validated by the standard FMVSS 208 of the National Highway Traffic Safety Administration (NTHSA) [1] with an innovative and simple method of measurement and operation. Without further, this article has the following structure: Sect. 2 shows the design of the prototype, materials, and essential components for its construction, Sect. 3 deals with the kinematic analysis of the mechanism and the physical phenomena present in the test bed, and Sect. 4 presents the experimental tests corroborating the correct operation by means of a video metric measurement. A brief conclusion is presented in Sect. 5.

2 Prototype Design The impact theory stipulates three types of impacts [3], present in a collision. • Primary—The vehicle contacts the object and the passenger compartment deformation begins. • Secondary—The users are projected with the cabin contacting the board, steering wheel, doors, among others. • Tertiary—This happens inside the body of the users, where the organs impact with the internal walls of the human body. Due to this phenomenon, the automobile, the user, and the internal organs have the same movement speed, so the kinematic energy study is required. This phenomenon is expressed by the next equation (Eq. 1). Ec =

1 2 mv 2

(1)

where Ec is the kinetic energy of the system, m represents the mass of the system, and v is the speed. Equation 1 allows to calculate the energy that is linked to the body in movement, where it is observed that the kinetic energy of a body in motion depends on the speed in the logarithmic form (which quadruples the energy). However, the collision to be considered in this work contemplates the secondary collision where the user impacts inside the vehicle’s interior. To know what the possible damage or the casualty possibility could be it is important to obtain a parameter that can measure injuries. To get this parameter, the HIC is used which helps to evaluate the automobile retention systems. Therefore, a kinematic value is necessary to calculate this parameter, acceleration. The acceleration provides all the compendium of energy dissipation because strain environment of the vehicle and the user depends directly with this value and the time. All these parameters are validated by the Standard FMVSS 208 of NTHSA.

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It is important to validate this experimental test by the standard. The impact tests of the New Car Assessment Program (NCAP) are standardized for validation in the case of a collision. This body establishes that the adequate speed to make a frontal impact is 30 mi/h ≈ 48.3 km/h ≈ 13.4 m/s [4]. Another design characteristic is the test bench dimensions; due to the reduce area that may occupy this mechanism, it is important to manufacture a small structure. For this reason, 2 m high will be the maximum length. This value is too important because it depends on how is going to get the energy necessary to accelerate the prototype to the speed established. To solve this dilemma, the gravity force is used as a speed generator in free fall. Nevertheless, the gravity force is not enough to get the desired speed, so a system of elastic bands is implemented to complement the total generation of energy. As this test bench is required to get by low cost, the dimensions play an important role in reducing the cost. The use of videogrametry was essential since it avoids the use of high-cost sensors and accelerometers, in addition to the data acquisition systems needed to read the impact values. In this way, a vertical platform with an internal mobile structure is proposed, which will impact the lower part of the bank against a non-deformable barrier. Figure 1 shows a schematic figure of the test bench and its dimensions. The previous design of the mechanism was developed on the SOLIDWORKS® software where the mobility of the mobile structure is seen in detail and allowed to create the construction plans of the mechanism (see Fig. 2).

Fig. 1 a Schematic the test bench and b test bench dimensions

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A. Cuautle-Estrada et al.

Fig. 2 a Start of the trajectory, b impact of the head, and c the height reached in the first bounce

3 Kinematic Analysis For the kinematics of the experiment, specifically final speed and acceleration of the proposed mechanism, there were used concepts that describe the behavior of dynamic loads such as Newton’s second law, the uniformly accelerated rectilinear motion, the potential energy and the Hook’s law for the calculation of the elastic coefficient [5]. The necessary variables were taken from the characteristics of the test bench design for the kinematic calculation as shown in Table 1. Table 1 Test bench variables

Variable

Value

g gravity

9.81 m/s

h height

2m

m mass

4.5 kg

Simplified Test Bench Used to Reproduce Child Facial … Table 2 Analytical calculation of the test bench

27

Concept

Equation

Free fall speed

vf 2

=

vi 2

Results + 2a X

v f = 6.26 m/s

Total mechanical work

UT = Uvdesired − Uvgravity

U T = 187.92 J

Elastic coefficient

K =

K = 588.6 N/m

Elastic potential energy

U=

Elastic band stretch length

Lt = L beginin+L elastic

Lt = 1.67 m

Potential energy

ε P = mgh

ε P = 404.01 J

F X 1 2 2 K (X )

X = 0.8 m

With these values, it is proceeding to the kinematic analysis of the mechanism. Table 2 shows the results obtained. As can be seen in Table 2, the most important results for the execution of the tests are shown. The total stretch length of the elastic bands is the most important data, and it is very important since it depends on this data that the desired final speed is reached.

4 Experimental Tests To carry out the experimental tests, the test bench should be positioned in an illuminated place, where the behavior of the impact can be appreciated in detail. Rail lubrication must be constant in all tests; also, the structure mobile weights the same in all tests. A digital camera of excellent resolution is necessary to be able to record the test since the measurement will be done with videogrametry. A Go Pro Hero 3 silver camera of 11 megapixels was used for this task [6]. The configuration used in the camera was 720p at 120 frames per second. As a final point, the mobile structure will start the test at 1.67 meters height, and it will be repeated at least 20 times to validate the results repetition. Figure 2 shows the experimental test. As can be seen in Fig. 2, the record has the main impact and a rebound. This rebound is due to the phenomenon of restitution [7], which is present in all bodies’ impact.

5 Results After having carried out the experimental tests and with the use of the Kinovea software, the speed and acceleration graphs are obtained. Figure 3 shows both speed and acceleration system.

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Fig. 3 Right, speed graph, left, acceleration graph

The speed reached was 13.4 m/s, which validates the test due to the speed established by the FMVSS 208 standard. In this way, the acceleration in the chart could be considered an accurate result, where the acceleration was 483 m/s2 in only 220 ms. This amount of acceleration dissipated in a short period of time and is responsible for causing the principal injuries to the child’s head.

6 Conclusion The magnitude of the severity of the head injury will be directly governed by the dissipation time of the energy. In addition, because the dynamic impact is sudden, the deformations and therefore the injuries are multiplied by a dynamic coefficient of impact, which will be obtained in future work with the calculation of the HIC. Nevertheless, this mechanism works appropriately simulating specific parameters of the Standard FMVSS 208 which specify to determine tests as valid, reaching the specific speed before the impact and the non-deformable barrier where the head crashes. Acknowledgements The authors thank for the support to National Polytechnic Institute (IPN) and National Council for Science and Technology (CONACYT) to make this work possible and the participation of the biomechanics group of INSIA incorporated to the Polytechnic University of Madrid (UPM), from Spain. The authors also thank the support of projects 1931 and 20196710, as well as an EDI grant, all by SIP/IPN.

References 1. Li, Q., et al.: Potential gains in life expectancy by improving road safety in China. Public Health 144, S57–S61 (2017)

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2. Hollowell, W.T., Gabler, H.C., Stucki, S.L., Summers, S., Hackney, J.R.: Updated review of potential test procedures for FMVSS No. 208. NHTSA Docket, 6407-6 (1999) 3. Jouvencel, M.R.: Latigazo cervical y colisiones a baja velocidad. Ediciones Díaz de Santos (2003) 4. Kullgren, A., Lie, A., Tingvall, C.: Comparison between Euro NCAP test results and real-world crash data. Traffic Inj. Prev. 11(6), 587–593 (2010) 5. Giancoli, D.C., Olguín, V.C.: Física: principios con aplicaciones (2006) 6. Frutos, J.B., Palao, J.M.: El uso de la videografía y software de análisis del movimiento para el estudio de la técnica deportiva. Lecturas: Educación Física y Deportes 17(169), 1–14 (2012) 7. García, O.: Energía disponible y restitución durante la collision (2006)

Scientific Visualization as a Tool for Signal Processing in EEG Interpretation: Car Driver Sleeping State Detection for Advanced Cars Control Systems Ivica Kuzmani´c, Igor Vujovi´c, Joško Šoda and Maja Rogi´c Vidakovi´c Abstract This paper integrates two scientific areas that are medical brain research into engineering, i.e., control engineering in traffic situations in order that by using scientific tools such as MATLAB and visualization for drowsiness detection of drivers. Scientific visualization is a tool that is used to present the basic idea of the proposed algorithm. The final goal of this research is to detect the driver’s sleeping state, which then can be used as an algorithm in autonomous system in cars as a help to prevent car accidents. Such an algorithm would be a trigger to the vehicle computer, and if an initial stage of sleeping is detected, the vehicle’s computer should override manual control and stop the car as soon as possible. A similar system can be used aboard ships, trains or airplanes. Research of this topic results in big data, which could be hard to understand without proper visualization techniques.

1 Introduction Automated car, truck or bus driving is a scope of many researches in science as well as in industry nowadays. Knowing human nature, it will always be old-fashioned people, who will like a human driver to seat at the driver’s seat and drive the car/bus/truck or any other transportation vehicle. Also, it has to be pointed out that every automation I. Kuzmani´c (B) · I. Vujovi´c · J. Šoda Faculty of Maritime Studies, Signal Processing, Analysis and Advanced Diagnostics Research - Boškovi´ca 37, 21000 Split, and Education Laboratory (SPAADREL), University of Split, Rudera Croatia e-mail: [email protected] I. Vujovi´c e-mail: [email protected] J. Šoda e-mail: [email protected] M. R. Vidakovi´c Laboratory for Human and Experimental Neurophysiology (LAHEN), Department of Neuroscience, School of Medicine, University of Split, Šoltanska 2, 21000 Split, Croatia © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_4

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system has turned off the switch to switch off automation and turn to manual controls, at least for now. Due to the lack of drivers in the labor market, same person needs to spend more hours in a vehicle which can lead that human driver will possibly be more tired, unrest or sleepy. In order to increase traffic safety, it would be necessary to integrate a system which will monitor the states of a driver when a vehicle is in “manual” operating mode. Some systems are at various stages of research, and some can be even found on the market in high-price cars. In this paper, we propose drowsiness detection based on brain–machine interface (BMI). Our EEG experience and original signals are based on [1, 2]. Our scope is only the drowsiness detection, but the input signals could be used for many other processes, i.e., for vehicle control by thoughts. This system would use optic or wireless (not preferable due to radiation) communication and could be coded similar to [3]. Brain– computer interface (BCI) was tested in [4] for the usage in virtual cars. Furthermore, EEG signals were used for BMI to help chronic stroke patients in movements by detecting motion intents [5]. According to [6], there are 100,000 car crashes as a direct result of driver fatigue and 1550 deaths every year. “National Sleep Foundation (NSF) found that drowsiness causes up to 1.9 million crashes a year, with 54% of all drivers having driven while drowsy at least once in the past year… [6].” The authors [6] proposed drowsiness detection using support vector machines. To use such a method, one must select a compact set of features, which enables differentiability between drowsy and awake state. They claim that the SVM classifier provided 97.94% accuracy. Another paper [7] used Android application to detect drowsiness, which is acceptable for practical implementation. It is implemented by minimal neural network structure obtained by compression of a heavy baseline model to a lightweight model. Number of car accidents caused by drowsiness is a real motive for our work. Due to the nature of the problem, this paper is multidisciplinary, including visualization techniques to confirm conclusions which are obtained by the signal processing. Visualization is also used in signal analysis. To make conclusions, it was necessary to consult neuroscience. Numeric calculation is under the signal processing and data acquisition based on electric or magnetic fields. Finally, traffic science is important to fit the research results into a realistic framework. This paper is organized as follows. The second section deals with the scientific visualization of M/EEG data. The visualization is necessary in the research of the topic to find a proper algorithm for the drowsiness detection, but it is not so important in actual operation of the proposed system in cars. The third section presents the proposed algorithm for drowsiness detection. Finally, conclusions are given in the last section.

2 Scientific Visualization It is well known that scientific visualization is must-have tool to interpret, describe and show results in appropriate form. Also, scientific visualization depends on tools and purposes. Nowadays, there are a lot of tools developed for various environments

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33

to visualize results, such as LATEX and Tikz [8] that is probably the most complex and powerful tool to create graphic elements and insert images. Also, MATLAB, Auto CAD, CATIA and SolidWorks to name it few are development platforms that are appropriate for research and visualization. Nowadays, very popular scientific tool is to use Python, a powerful stand-alone platform with integrated vector graphics for scientific visualization. For example, in [9], Python server is used, and script is developed to clean and visualize EEG data. The NeuroExplore application is developed, with roots in Python, which serves for visualization of brain patterns. Python can be used for visualization in many word processing applications, which is suitable for scientific publishing. An introduction to processing and visualization of MEG and EEG in Python is addressed in [10]. There is also a tool Plotly in Python, which is specialized in processing and visualization of MEG/EEG data [11]. EEGNET is presented in [12], as a tool that can be used for analysis and visualization of functional brain networks from MEG or EEG recordings. It is running under MATLAB (which has a new strategy to spread out in the scientific community by TAH licenses). SLEEPNET [13] is a similar tool, which is run by a recurrent NN, but it is a self-standing application, not connected to a MATLAB or Python. In our work, the processing, analysis and visualization are performed in MATLAB. In order to visualize data, it is necessary to develop visualization procedures. MATLAB offers a set of useful tools for signal analysis and visualization, why is MATLAB chosen for research.

3 Neurosicence Background and Signal Processing Approach to EEG Analysis Standard tools for EEG signal processing are, besides classical FT, empirical mode decomposition [14] and wavelets [15]. During the time, several enhancements were used, such as Q-factored wavelets [16] and wavelet neural networks [17]. Normal sleep is described in [18] based on EEG signal. An empirical model for tracking sleep process is developed in [19]. A convolutional neural network (CNN) is trained to automatically extract features from raw 30-second epochs of the EEG, EMG and EOG signals in [20]. A deep learning sleep stage classification is elaborated and presented in [21], and for feature extraction an epoch of 30 s was used. However, in reference [22], it is suggested that epochs of 30 s long might not be enough to detect sleep state. An epoch that is 60 s long might be a more suitable choice to perform feature extraction and processing to detect sleep state. A lot of references about sleep classification, sleep disorders and sleep stages detection show that this is a field with high potential of research, and, consequently technical applications. However, usually sleep state [22] is sometimes underestimated as something which is self-understood. On the contrary, sleep state is hard to exactly

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determine, because it is not only determined by the status of non-rapid eye movement sleep (NREM) and rapid eye movement (REM) sleep stages, but also with transient activation phases. The example of how hard it is can be the fact: “The onset of drowsiness is characterized by gradual or brisk “alpha dropout”.” When dealing with detecting drowsiness in drivers, it is possible to focus on alpha dropout detection, because early detection is better than to be too late and cause the car accident. It is well known that alpha waves usually cover range between 8 and 13 [Hz]. Drop of the magnitude in that range indicates so-called alpha dropout. Hence, the decrease of the magnitude is a measure that could be investigated in order to determine alpha dropout. But it must be pointed out that this dropout measure perhaps will not be sufficient for reliable detection because α-band is not present in all human beings. Several types and factors make influence on alpha dropout detections, namely there are other “types” of human beings that produce different waves in range of α-band suspension or total blockade. Also, some people change the range of α-waves when dropping to sleep which results that this group could not be easily detected. For such type, it is important to have fine tuning of the detection mechanism in order to establish referent frequency range for comparison with current frequency range. Otherwise, it is hard to distinguish α-band from other only using spectral analysis. Brain EEG/EMG signal processing and analysis are very complicated due to several issues that occur. One of the basic problems is the instability of sampling in medical instruments that are used to obtain signals. For example, if the stated sampling rate is 4 kHz, actually it can be 3.7 or 4.1 kHz. Since there is no reliable information on the sampling rate, any component of the signal can be known only approximately, and it would be prudent to have more signals, and to use a statistical expectation as a measure to estimate sampling rate. However, in the practice, there is only one recording of the same patient at the same conditions. Another problem is the noise. Noise varies from the conditions in the patient’s environment. There are also various electrical internal/external noise sources or equipment noise source. Various electric/electronic devices induce the noise as well. Finally, noise can be generated inside human body. Some of the noises are additive, and some multiplicative, some can be estimated using statistical measures some do not. Usually, good ratio between signal and noise (S/N ratio) is desirable. Generally, if the noise produces random phase, it is too high to restore the origin signal. There are several phases of the signal processing and analysis in case of the brain signals as follows: • • • •

Cutting time-line signal, DC offset removal, Hanning windowing and Spectral (or EMD, time–frequency, etc.) analysis.

Depending on the scope of the research, brain signals can be cut into small epochs, i.e., 250 [ms] or greater. In the case of sleeping, epochs would be between 30 and 45 s. Some empirical experiences suggested that even longer window would be more appropriate. On the other side, bigger epochs require more processing time which can cause a car accident. Manual or automatic cutting signal in time domain

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35

is actual implementation of windowed transformation. If the window is sufficiently small, it would not matter that the EEG signal is non-stationary signal. All of these factors show complexity of choosing the right window, which could be a scope of one or more articles.

4 Proposal It is very hard to determine the beginning of the sleeping from the EEG/EMG. Even experienced personnel can disagree about that in specific patient. However, for engineering purposes, it is not even necessary to detect actual sleeping or presleeping stage. It is merely enough to suspect the beginning of the (pre)sleep stage. Engineering solution is to ask the human subject whether he/she is awake or not. Based on the previous section and/or [22], we can simplify the procedure of the detection by monitoring only α-waves (frequency range from 8 to 13 Hz). When relative amplitude drops out, the suspected event occurred. However, since this method greatly depends on many factors, such as sex (cycles in woman), age (α shifts during years), α-blockade and α-suspension, it is imperative to take as many factors as possible to be more accurate. Although we used standard array to record EEG signals, a novel BMI should be developed in commercial solution. Following stages are signal processing (in general could be WT, EMD or FFT) and α-features detection. Since, it is not enough to determine if someone is asleep, additional technical solution is provided: The driver has to click the button when the computer asks if he/she is awake. If the button is not pressed, the vehicle deaccelerates and stops by the procedure for stopping. The proposal is shown as the algorithm in Fig. 1. In theory, the proposed algorithm should work. However, there is a detail which could become an obstacle in any algorithm. This is the problem of the decision: How much of decrease in amplitude, i.e., threshold, could be tolerated? (in other case that would be false alarm). This is usually solved with thresholding operation. If the values are below the threshold, then there was not the dropout. If the change is so big, i.e., passes the threshold, then it is actual detection. Even this is not so simple. How to determine the threshold and threshold of what are the questions that must be solved? A safe way to measure the threshold is to use the energy or power of the spectrum. Since every person has different brain patterns, the same person must be used for the reference and for the detection of the dropout. Algorithm for α-dropout detection (block detected α-dropout in Fig. 1): • Taking first several seconds to establish reference average brains signal energy • Calculating new average every second (current energy) • If the current energy is lower than threshold (in percentage of reference energy), then that the dropout occurs.

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Fig. 1 Proposed algorithm

5 Results In order to show how the proposed algorithm works, we have generated an EEG signal. In order to generate the signal, we firstly generated the first second with 3000 Hz sampling:

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37

⎧ sin(2π 21 t) + sin(2π · 2t) + sin(2π · 9t) + sin(2π · 11t) ⎪ ⎪ ⎪ ⎪ ⎪ + sin(2π · 35t) + sin(2π · 80t) for t < 1/3 [s] ⎪ ⎪ ⎪ ⎪ 2 · sin(2π 21 t) + 0.5 · sin(2π · 2t) + 3 · sin(2π · 9t) ⎪ ⎪ ⎨ +0.9 sin(2π · 11t) + sin(2π · 35t) brain_signal = (1) ⎪ + sin(2π · 80t) for 1/3 < t < 2/3 [s] ⎪ ⎪ ⎪ 1 ⎪ ⎪ 0.5 · sin(2π · 2 t) + 1.5 · sin(2π · 2t) + 0.4 · sin(2π · 9t) ⎪ ⎪ ⎪ + 0.9 · sin(2π · 11t) + sin(2π · 35t) ⎪ ⎪ ⎩ + sin(2π · 80t) for 2/3 < t < 1 [s] ⎧ sin(2π 21 t) + sin(2π · 2t) + sin(2π · 9t) + sin(2π · 11t) ⎪ ⎪ ⎪ ⎪ ⎪ + sin(2π · 35t) + sin(2π · 80t) for t < 1/3 [s] ⎪ ⎪ ⎪ ⎪ sin(2π 21 t) + sin(2π · 2t) + sin(2π · 35t) ⎪ ⎪ ⎨ + sin(2π · 80t) for 1/3 < t < 2/3 [s] brain_alpha_dropout = ⎪ sin(2π 21 t) + sin(2π · 2t) + 0.01 sin(2π · 9t) ⎪ ⎪ ⎪ ⎪ + sin(2π · 35t) + sin(2π · 80t) for 2/3 < t < 0.833 [s] ⎪ ⎪ ⎪ ⎪ sin(2π 21 t) + sin(2π · 2t) + 0.1 sin(2π · 9t) ⎪ ⎪ ⎩ + sin(2π · 35t) + sin(2π · 80t) for 0.833 < t < 1 [s] (2) Then, the simulated signal is normalized. To create more cycles, we used vector notation in the MATLAB: br _sig_total = [brain_sig . . . brain_sig . . . brain_sig . . . brain_sig_dr op . . . brain_sig . . . 0]

(3)

This simulated the input signal to the proposed algorithm. Figure 2 shows the timedomain representation of brain signal with and without α-dropout. It is obvious that we cannot make a conclusion from this graph easily. Hence, some sort of scientific visualization is necessary. One of the ways is to use signal processing to transform data into more appropriate domain. In this case, windowed FT is used to transform the brain signal. However, FT is constrained with natural time division based on the signal nature, and it is not based on math for STFT. It is primitive visualization. In order to use advanced graphs, 3D visualization or similar, the nature of the signal should be changed, which is not realistic. In order to operate, the algorithm does not need visualization. It is necessary for scientists to make conclusions that would be used in the algorithm.

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Fig. 2 Voltage versus time graph for brain signal with and without alpha dropout in one-second window

Figure 3 shows spectrum of the brain signal in case of awake person and in case of α-dropout. It is observable that magnitude exhibits a significant difference. However, the phase component of the spectrum is not so straightforward. Figure 4 illustrates how even a simple 2D plot can be useful to distinguish awake and asleep persons by the α-dropout feature. Figure 5 shows that even 3D rotation does not always produce better results. It is an example where 3D does not help. The same is easily seen from a simple 2D plot. 3D view could not be obtained, because function such as mesh, surf or 3D plot needs 3D data. Hence, some other technique must be used. Next step was to use Signal Analyzer app. It was expected that professional visualization would produce better results. The result of some basic options is shown in Figs. 6 and 7. Figure 8 shows the results of normalization in time domain. It is obvious that the results are not spectacular.

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Fig. 3 Visualization of frequency information in windowed FT by MATLAB

6 Conclusions To be able to do research and development, a visualization of measured data is of paramount importance. We have used MATLAB as a scientific visualization tool with example of EEG brain signal to estimate drowsiness in driver using alpha dropout as a threshold measure. From obtained data, it can be seen that MATLAB is a very useful tool for initial and breakthrough research, but platform-dependent for serious

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Fig. 4 Comparison of spectra for the two cases in 2D plot using stem function in MATLAB

Fig. 5 Example when 3D visualization does not help

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Fig. 6 Example of Signal Analyzer app usage: a spectra, b sampled time and normalized frequency

independent applications. Our research and interpretation of measured data are based on visualization in time domain and frequency domain in 2D and 3D spaces where some patterns and trends are shown. In order to develop accurate and reliable estimates, both domains are important for feature extraction and interpretation because in each domain we add important pieces to our knowledge of the investigated prob-

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Fig. 7 Result of the low-pass filter application

Fig. 8 Normalized y-axis in t-domain (it can be seen that it is not so convenient view)

lem. Also, some other platforms are a better choice for product development such as Python, Visual C, C# and Java. There is a need for a practical data acquisition instead of cup with electrodes, i.e., electrodes positioned at the weal or the driver’s seat.

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Acknowledgements This work was performed in cooperation with the following scientific groups “Applied Signal Analysis of Biomedical Signal’s” and “New technologies in maritime” established at The University of Split, Faculty of Maritime Studies.

References 1. Vidakovi´c, M.R., Jerlovi´c, A., Juri´c, T., Vujovi´c, I., Šoda, J., Erceg, N., Bubi´c, A., Schönwald, M.Z., Liumis, P., Gabelica, D., Ðogaš, Z.: Neurophysiologic markers of primary motor cortex for laryngeal muscles and premotor cortex in caudal opercular part of inferior frontal gyrus investigated in motor speech disorder: a navigated transcranial magnetic stimulation (TMS) study. Cogn. Process. 17, 429–442 (2016) 2. Vidakovi´c, M.R., Gabelica, D., Vujovi´c, I., Šoda, J., Batarelo, N., Džimbeg, A., Schönwald, M.Z., Rotim, K., Ðogaš, Z.: A novel approach for monitoring writing interferences during navigated transcranial magnetic stimulation mappings of writing related cortical areas. J. Neurosci. Meth. 255, 139–150 (2015) 3. Mazi´c, I., Bonkovi´c, M., Bjelopera, A.: The Manchester coded data based OFDM (MCOFDM). Trans. Marit. Sci. 7(2), 154–163 (2018) 4. Wang, H., Li, T., Bezerianos, A., Huang, H., He, Y., Chen, P.: The control of a virtual automatic car based on multiple patterns of motor imagery BCI. Med. Biol. Eng. Comput. 57(1), 299–309 (2019) 5. Bhagat, N.A., Venkatakrishnan, A., Abibullaev, B., Artz, E.J., Yozbatiran, N., Blank, A.A., French, J., Karmonik, C., Grossman, R.G., O’Malley, M.K., Francisco, G.E., Contreras-Vidal, J.L.: Design and optimization of an EEG-based brain machine interface (BMI) to an upper-limb exoskeleton for stroke survivors. Front. Neurosci. 10, 122 (2016) 6. Yu, S.: EEG-based drowsiness detection using support vector machine. M.Sc. Thesis. Graduate and Professional Studies of Texas A&M University (2014) 7. Jabbar, R., Al-Khalifa, K., Kharbeche, M., Alhajyaseen, W., Jafari, M., Jiang, S.: Real-time driver drowsiness detection for android application using deep neural networks techniques. Procedia Comput. Sci. 130, 400–407 (2018) 8. TikZ package. https://www.overleaf.com/learn/latex/TikZ_package. Accessed 22 Dec 2018 9. dos Santos Rocha DJ (2017) NeuroExplore—visualizing brain patterns—a physiological computing InfoVis. M.Sc. Thesis, Técnico Lisboa 10. Gramfort, A., Larson, E., Luessi, M., Engemann, D., Brodbeck, C.M., Hämäläinen, M.: Intro to MEG and EEG processing with MNE and Python. https://mne-tools.github.io/mne-pythonintro/. 30 June 2018 11. Process MEG/EEG Data with Plotly in Python. https://plot.ly/ipython-notebooks/mne-tutorial/. Accessed 15 Nov 2018 12. Hassan, M., Shamas, M., Khalil, M., El Falou, W., Wendling, F.: EEGNET: an open source tool for analyzing and visualizing M/EEG connectome. PLoS ONE 10(9), e0138297 (2015) 13. Biswal, S., Kulas, J., Sun, H., Goparaju, B., Westover, M.B., Bianchi, M.T., Sun, J.: SLEEPNET: automated sleep staging system via deep learning. arXiv:1707.08262v1 [cs.LG] (2017) 14. Huang, N.E., Attoh-Okine, N.O.: The Hilbert-Huang transform in engineering. CRC Taylor & Francis (2005) 15. Vujovi´c, I., Šoda, J., Kuzmani´c, I.: Cutting-edge mathematical tools in processing and analysis of signals in marine and navy. Trans. Marit. Sci. 1(1), 35–48 (2012) 16. Hassan, A.R., Siuly, S., Zhang, Y.: Epileptic seizure detection in EEG signals using tunable-Q factor wavelet transform and bootstrap aggregating. Comput. Methods Programs Biomed. 137, 247–259 (2016) 17. Zaniol, C., Varriale, M.C., Manica, E.: Apnea recognition with wavelet neural networks. Tend. Mat. Apl. Comput. 19(2), 277–288 (2018)

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18. Benbadis, S.R.: Normal sleep EEG. Medscape. https://emedicine.medscape.com/article/ 1140322-overview (2018). Accessed 4 Apr 2018 19. Prerau, M.J., Hartnack, K.E., Obregon-Henao, G., Sampson, A., Merlino, M., Gannon, K., Bianchi, M.T., Ellenbogen, J.M., Purdon, P.L.: Tracking the sleep onset process: an empirical model of behavioral and physiological dynamics. PLoS Comput. Biol. 10(10), e1003866 (2014) 20. Kern, S.J.: Automatic sleep stage classification using convolutional neural networks with long short-term memory. M.Sc. Thesis, Radboud University, Nijmegen (2017) 21. Chambon, S., Galtier, M., Arnal, P., Wainrib, G., Gramfort, A.: A deep learning architecture for temporal sleep stage classification using multivariate and multimodal time series. IEEE Trans. Neural Syst. Rehabil. Eng. 26(4), 17683810 (2018) 22. Scomer, D.L., Lopes da Silva, F.H.: Niedermeyer’s electroencephalography—basic principles, clinical applications and related fields, 6th edn. Wolters Kluwer, Lippincott Williams & Wilkins, Philadelphia-Baltimore-New York-London-Buenos Aires-Hogn Hong- Sydney-Tokio (2011)

Determination, Validation, and Dynamic Analysis of an Off-Road Chassis Mateus Coutinho de Moraes and Miguel Ângelo Menezes

Abstract The chassis of a car can be interpreted as a large spring connecting the rear and the front suspension. A flexible chassis adds another spring to an already complex system being difficult to control the handling and especially the vehicle lateral load transfer. Factors such as handling as well as vibrations are primordial, since a more rigid structure implies smaller prototype deformation along the path. The torsional stiffness relates to the torsional deflection of the structure when subjected to a pure torque acting on the vehicle’s longitudinal axis. The torsional stiffness is one of the main design criteria of a chassis. The problem of settle the best relationship between cost, weight, and chassis rigidity is of great importance in both the development of popular and high-performance cars. The influence of the chassis torsional stiffness on the dynamic and kinematic suspension behavior is still poorly explored, and there are only a few studies in the literature referred for Baja SAE vehicles. This work has the aim of exposing the boundary conditions necessary to determine the chassis torsional stiffness based on the finite element method. Furthermore, the explanation of the methodology to be used for the determination of this parameter in practice and finally the post-processing in which are used lateral dynamic concepts to assess how the chassis torsional stiffness influences the dynamic and kinematic behavior of the vehicle.

1 Introduction The classical analytical methods allow the exact calculation of the displacements, deformations, and tensions in a structure at all its points. This is only possible in simple geometries ordered by very well-defined loads. Thus, the finite element method arises to supply the need to solve these problems for more complex geometry and boundary conditions, within the acceptable precision of the engineering problem. M. C. de Moraes · M. Â. Menezes (B) Ilha Solteira Engineering Faculty, UNESP—São Paulo State University, DEM, Av. Brasil, 56, Centro, Ilha Solteira 15385-000, SP, Brazil e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_5

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The finite element method is an approximate method of calculating continuous systems in which the continuous body is subdivided into a finite number of parts or elements, the nodes. A finite number of parameters specify the mathematical model. In problems of structural analysis, the unknowns and parameters are the nodal displacements. The fundamental laws in the design of the mathematical model that represents the discretized structure are the balance of forces, compatibility of displacements, and the material behavior law. The simultaneous behaviors of bending actions and forces in the plane of the plate make extensive use of shell elements as detailed analysis of chassis and sleepers, trains, and airplanes [1]. Chassis is defined as the basic structure of a motor vehicle in which it is built. It has the function of containing, supporting, and connecting the other vehicle parts. Ideally, the main purpose of the car chassis is to connect the four wheels with a rigid structure to bending and twisting. This has to support all components and must absorb all loads without deflection. In addition, it should connect the suspension systems, protect the pilot from accidents, transmit workloads, and accommodate powertrain components, among other auxiliary systems [2]. The chassis torsional stiffness is a very important information because it relates to vehicle dynamic responses and is usually modeled when connecting the axles through a torsion spring. Increasing the chassis torsional stiffness of a racing car optimizes handling with the optimum adjustment being the one where the roll stiffness originates only from the suspension [3]. Factors such as drivability as well as vibrations are paramount, since a more rigid structure implies smaller vehicle deformations throughout the design. The problem is the determination of the best relationship between chassis stiffness, weight, and cost, which is of great importance in both the development of high performance and popular vehicles (cost efficiency and consumption). The maneuverability and vibrational behavior of a vehicle are directly influenced by the chassis stiffness. The torsional stiffness relates the structure torsional deflection when subjected to a pure torque acting on the vehicle longitudinal axis. In car competitions, the main factors related to torsional stiffness are lateral load transfer, dynamic, and kinematic aspects of suspension [4]. The chassis torsional stiffness can be measured in several ways. Each mode consists of the basic principle of attaching one of the chassis ends and applying a torque to another in such a way that torsion occurs and is measured [5]. A typical test involves the chassis twisting, measuring its angular variation, and then returning it to the starting point. For validation aiding, the value obtained is compared to the stiffness of the calculated structure in an analytical model [3]. Thus, the torsion angle is measured for a respective torsion moment. Figure 1 shows a representative scheme of the torsional stiffness concept. The test benches can have three configurations, in each one supports have a certain arrangement. In all arrangements, the principle is keeping the rear fixed when a torque is applied to the front [6]. In the center pivot configuration, there are two fixed supports in the rear and the third point is in the front center; corner pivot is just as in the previous case, and there are two fixed supports in the rear; however, the

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Fig. 1 Representative scheme of the concept of torsional stiffness. From [3]

pivot is now in the corner. For both previous arrangements, a vertical load outside the longitudinal center applies to the front to generate a torque in the chassis. The two rigid supports is the configuration that has four support points with the floor, and only the rear supports remain fixed. The front brackets generate vertical displacements such that a torque is generated. Steady-state conditions are constant velocity in the curve, as well as longitudinal acceleration and flat road, in other words, without irregularities. Also, all the data of the car used in calculations are linear (roll rates and spring stiffness coefficients), and basic data of dimensions (CG height and wheelbase) are constant. Thinking the chassis as a large spring that connects the front and rear suspensions, if the torsion spring (chassis) is weak, the attempts for controlling the distribution of the lateral load transfer will be at best confusing and impossible, at worst. This is because a flexible chassis adds another spring to an already complex system. Predictable handling can be better achieved if the chassis is rigid enough to be safely ignored. The goal of torsional stiffness is to provide a rigid platform for suspension, allowing lateral loads to be distributed from front to back in proportion to the suspension’s rolling stiffness [7]. The lateral load transfer is a function of the stiffness distribution for the suspended mass (rolling stiffness), unsuspended mass, as well as the position of the rolling axis. Lateral load distribution is typically modified by changing the height of the CG or

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Fig. 2 Non-suspended mass of a vehicle. From [9]

the track width, to change the clearances present on the wheel or when you use the wheel hub spacers. The load transfer has the contribution of the non-suspended mass as well as the suspended mass, which is subdivided into a portion related to the lateral force and another connected to the torsion angle [8]. The non-suspended mass includes unsupported suspension components, such as wheels, bumpers, and the brake subsystem, as illustrated in Fig. 2. The lateral load distribution of the suspended mass is composed of lateral force components caused by the steering. They act on the roll center (points where there is a coupling between forces acting on the suspended and the non-suspended mass) of each axis, and these are determined by the suspension kinematics, so this part is known as kinematic component. The other fraction relates to the centrifugal inertia force that generates a moment causing the suspended mass to roll to the outside of the curve. When this happens, the outer spring of the suspension compresses while the inner part extends. The forces in the springs have a reaction, which is the elastic component of the load distribution. As the chassis rolls, the center of gravity of the suspended mass shifts to the side, creating a moment that increases side load transfer. The torque that the suspension makes when the body rolls trying to put it back in its original position at the angle of rotation of the vehicle around a longitudinal axis defines the roll stiffness. This factor is directly related to the weight distribution, and its adjustment can be made with the use of stabilizing bars. The terms influencing the distribution of the lateral load transfer are the fraction of the rolling time (rolling moment) generated by the suspended mass and resistance of the front axle, force applied to the axle beyond the effect of the unsuspended mass. Rolling (φ) is visible when the vehicle performs curves, as shown in Fig. 3. According to [7], when comparing the front stiffness and rear axles with the chassis torsional stiffness, it should resist approximately the difference between the distribution rates between the front and rear rotation rates. However, recent studies have shown that there is proportionality between suspension roll stiffness and chassis’ torsional stiffness, a multiple number that varies with each design [11]. Bending stiffness is not as important as torsional stiffness. The change in performance is more significant when deformation occurs, twisting the vehicle. Therefore,

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Fig. 3 Vehicles steering, situation in which it is possible to notice the rolling of the chassis. From [10]

a number of problems arise due to the lack of stiffness of the chassis, such as difficulty in controlling the distribution of lateral load; lack of the maneuverability; vibrations; shifts in connection points of the chassis with the suspension, causing lack of tire control, and unstable behavior of the vehicle as a whole.

2 Methodology Finite element simulations are performed using the Ansys software provided by ESSS, and the equations that will allow the study have been implemented in MATLAB. In the paper, it will be evaluated the chassis lateral load distribution in models employed previously by the TEC Ilha Baja SAE team in 2016, 2017, and 2018 years. These three vehicles had the same suspension system and could be observed as the torsional stiffness of each chassis has influenced on lateral load distribution, allowing to estimate an optimum value for the suspension project. The aim is to implement this methodology in preliminary steps of the chassis design, which requires certain suspension project data that will be further better explained.

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2.1 Simulation For analyzes to be carried out on vehicle chassis, shell elements are generally used to model the tubes in order to obtain a more uniform mesh and facilitate the geometry processing during the simulation. In the process of converting the geometry to shell, the thicknesses (wall) of tubes are defined. The analysis consists of mounting the chassis at the rear suspension support points while applying binary of vertical forces (Y-axis only) at points of the front shock absorbers and then measuring the deformations at those points. The boundary conditions can be seen in Fig. 4. The measurements were making using the directional deformation in the X- and Y-directions, varying the forces from 100 up to 1000 N. So, through the results of the simulation, a graph torque x angular deformation was constructed. The torque and angular deformation have been calculated as follows: T = FL

(1)

and θ = tan−1

 y

m´in

   + |ym´ax |

180 L/π

(2)

where T = Torque [Nm]; L = Distance between the two applied forces [m]; θ = Angular deformation [º]; ym´ax = Maximum deformation in the Y-direction [m] and ym´in = Minimum deformation in the Y-direction [m].

Fig. 4 Boundary conditions used in the simulations

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2.2 Validation The torsional stiffness test consists in applying a binary force on the axis of the front suspension, and mounting the rear suspension in order to restrict its movement in all directions. Three supports are used for the test, as shown in Fig. 5. Also, two rigid supports are attached to the chassis rear and the floor preventing all degrees of freedom. On these supports, it is fixed a bar that passes between the rear semitrailing arms for locking the rear, in which are welded the two supports. The third support is placed in the front center. A force F is applied out the front center causing a known torque in the vehicle chassis. Thus, the torque generated results in a torsion angle. A support with weights is placed on the spindle tip of the front wheel. In each axis ends, a comparator clock with one-thousandth of a millimeter precision was placed to obtain the displacements. When the deformation occurs, a comparator clock measures how much it deforms upwards, and the other downwards, as shown in Fig. 6.

Fig. 5 Supports used in the experimental apparatus

Fig. 6 Support with weights and the comparator clock and the positioning of the comparator watch, respectively

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For the chassis torsional stiffness test, some car elements are removed, as transmission, electronic, brake, steering, and the suspension subsystems maintaining only the knuckles and the axle tip. To nullify the effect of car shock, absorbers were used rigid metalon, as shown in Fig. 7. The degree of freedom of the knuckles allowing movement of the wheel was compensating by tying up it to the suspension trays using steel wire. Due to material limitation, a maximum load of 156.96 N was applied. Four 4 kg masses were used. Thus, the maximum torque obtained was 98.1 Nm. The displacements have also been checked five times, in both masses loading and unloading. The final apparatus of the test is shown in Fig. 8. The formula used for obtaining the torque is: Tprat = F D

Fig. 7 Metalon replacing the shock absorbers and support for the weights, respectively

Fig. 8 Assembled test

(3)

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Being, D, the distance between the pivot points of the car to a comparator clock. For the angle, it has been: θ = tan

−1

 y

m´in

   + |ym´ax |

180 J/π

(4)

in which, J is the distance between the comparator clocks. With obtained data from the test, a graph of the torque by the deformation angle was plotted. The value of the angular coefficient of this plotting represents the experimental torsional stiffness.

2.3 Dynamic Analysis The methodology employed for determining the curve of the lateral load transfer percentage x roll stiffness in the front, as treated before [6], was based on the SAE paper [9]. Initially, the directional deformation values obtained by means of simulations for each model of chassis are inserted. Then, it is calculated the suspension roll stiffness. It is composed by the rolling of springs and anti-roll bars quantified by: K front_1 =

0.5K molafront z tfront a tan(z/tfront )

(5)

Wherein K molafront is the spring constant of the front spring. If are installed antiroll bars, their torsional rigidities should be added to the spring constant of springs to compose this term. z is the sum of the vertical deflections obtained by the simulation, and t front other fraction is the roll related to tires: K front_2 =

K pneufront z tfront a tan(z/tfront )

(6)

The term K pneufront refers to the radial stiffness of the front tire. This is a system composed of two springs in series, so the total rolling is: K suspfront =

K front_1 · K front_2 K front_1 + K front_2

(7)

The same procedure is applied to the rear suspension, being calculated the roll stiffness of damper springs and anti-roll bars, such as the rolling caused by tires and finally the total rear roll stiffness. For the calculation of the load transfer, it is needed to inform a set of data related to the car: the mass on the front and rear axle, the lateral acceleration as well as the height of the gravity center. The torsion angle of the front can be calculated using (8) and of the rear using (10), as follows:

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 ϕf = 

 K susprear Mfront + K chassi Mtot .al.hcg

K susprear K suspfront + K chassi K suspfront + K chassi K susprear   Mrear .al.hcg − K susprear ϕ f   ϕb = K susprear + K chassi ϕr = ϕb + ϕ f



(8) (9) (10)

The lateral load transfer on the front axle is: L LT f =

K suspfront ϕ f tfront

(11)

The same equation is utilized for the rear axle. For plotting the curve described in this method and which will be presented in the results, the roll stiffness distribution in the front axle varies in steps of 5%. It is calculated simultaneously the rear roll value, the torsion angles of the front, chassis and the rear, as well as the new lateral load distribution of the front axle. The literature presents a series of equations that allow estimating the lateral load transfer without considering the chassis torsional stiffness. So, the ideal distribution was based on [7], as follows: z=

h rcf + h rcr 2

(12)

Wherein hrcf is the roll center height, while hrcr is the height of the rear roll center. H = hcg − z

(13)

The lateral load transfer of the front axle is L L T f ideal =

H.K suspfront al.Mtot b.h rcf · + tfront (K suspfront + K susprear ) l

(14)

where the term b is the distance between the longitudinal center of gravity and the rear axle. b=

Mfront.l Mtot

(15)

And the term l is the wheelbase. Yet, the ideal lateral load transfer in the rear axle is expressed by; L L T rideal =

H.K susprear al.Mtot a.h rcr · + trear (K suspfront + K susprear ) l

(16)

Determination, Validation, and Dynamic Analysis of an Off-Road …

55

where the term a is the distance between the longitudinal center of gravity and the front axle. The literature shows a series of equation that allows estimating the lateral load transfer without considering the chassis torsional stiffness. So, the ideal distribution was based in [8].

3 Results and Discussions 3.1 Simulations The results of the sum of the deflection obtained by the simulations of each chassis configuration are presented in Table 1. For the simulation, the forces were varied from 100 to 1000 N and the maximum and minimum deformation values are used to generate the graph representing the torsional stiffness. The superposition of the straight, which allowed the calculation of each torsional stiffness, is presented as follows in Fig. 9. The chassis torsional stiffness is equal to 646.72 Nm/° for the 2016 model, 1185.59 Nm/° for 2017, and 1629.64 Nm/° for the one used in 2018.

3.2 Validation Regarding the experimental data, the mean displacements of five measurements are used. The deformations were measured by loading the first mass of 4 kg up to the fourth mass, and then discharging them, until all masses were removed. The distance Table 1 Chassis deformation values of the TEC Ilha Baja SAE team Force (N)

Deformation of 2016s chassis (m)

Deformation of 2017s chassis (m)

Deformation of 2018s chassis (m)

100

0.0005

0.0003

0.0002

200

0.001

0.0005

0.0004

300

0.0015

0.008

0.0006

400

0.002

0.001

0.0008

500

0.0025

0.0013

0.0009

600

0.003

0.0016

0.0011

700

0.0035

0.0018

0.0013

800

0.004

0.0021

0.0015

900

0.0045

0.0023

0.0017

1000

0.0049

0.0026

0.0019

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Fig. 9 Representative of straight torsional stiffness of each TEC Ilha Baja SAE team chassis (years of 2016, 2017, and 2018)

between the pivot point and the comparator clock was 62.5 cm, while between the two comparator clocks was 143 cm. The mean values of the experimental data collected are presented in Table 2. In this way, the product is made between the average angular coefficient of the five straight lines and the arm (distance from the pivot point to the comparator clock); hence, the experimental torsional stiffness is determined and equal to 640.29 Nm/º for the 2016 chassis. The comparison between theoretical values (simulations) and experimental values has indicated a percentage error of 1%. According to the literature [8], errors up to 20% would be acceptable, which proves the validity of the test performed. Figure 10 shows the representation of the data collected in the validation of the chassis of 2016. Table 2 Mean values of deformation obtained in the practice test

Force (N)

Mean deformation of left clock (mm)

Mean deformation of right clock (mm)

39.24

0.706

0.28

78.48

1.382

0.559

117.72

2.06

0.849

156.96

2.772

1.183

156.96

2.876

1.235

117.72

2.295

0.937

78.48

1.636

0.63

39.24

0.943

0.315

0

0.088

0.016

Determination, Validation, and Dynamic Analysis of an Off-Road … Table 3 Input data for modeling the curve of front-load transfer as % of total load transfer x front roll stiffness as % of total roll stiffness

57

Parameter

Value

Stiffness coefficient of the front spring [N/m]

10,500

Stiffness coefficient of the rear spring [N/m]

15,000

Axial rigidity of the tires (front and rear) [N/m]

60,195

Torsional stiffness of rear anti-roll bar [Nm/°]

673.85

Distance application binary of forces [m]

0.420

Front track width [m]

1.44

Rear track width [m]

1.34

Wheelbase [m]

1.45

Front roll center height [m]

0.1395

Real roll center height [m]

0.1347

Front mass [kg]

107.5

Rear mass [kg]

142.5

Lateral acceleration (m/s2 )

9.81

Center of gravity height [m]

0.588

Fig. 10 Straight line representative of the torsional stiffness of 2016s chassis

3.3 Dynamic Analysis The following data of the car were used to model the curve of lateral load transfer on the front axle x percentage of roll stiffness on the front axle. The curve obtained is presented as follows in Fig. 11: The deflections in the Y-direction were also used to evaluate the suspension roll stiffness, and the values obtained were 174.76 Nm/º for the front and 217.31 Nm/° for the rear, totaling a suspension roll stiffness equal to 392.07 Nm/°. That way the distribution of rolling is 44.39:56.61. With the deflections and other data of the car

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Fig. 11 Front-load transfer as % of total load transfer x front roll stiffness as % of total roll stiffness for vehicles of TEC Ilha Baja SAE team used in 2016, 2017, 2018 years as well as the ideal based in [8]

suspension, the height of the center of gravity and the distribution of the mass were possible to estimate the lateral load for each axle for each chassis. These values are presented in Table 4. The analysis of data presented in Table 3 as well as the values generated by the program shows that the increasing of chassis torsional stiffness means there is a decreasing of load on the front and rear axles. The increase in the relationship between the chassis torsional stiffness and the suspension roll stiffness caused the lateral load transfer to become loser to ideal. In addition, the decrease in load transfer values on each axis is beneficial because it improves traction. The search for minimization of this parameter is only possible when the chassis is assumed to have an infinite torsional stiffness, so it should be sought to obtain a decrease in the error of the lateral load distribution without implying an excessive increase in mass. It should also be kept in mind that Baja vehicles are oversteer, and so they should not have excessive traction. Table 4 Calculated parameters Chassis

Torsional stiffness [Nm/°]

Lateral load distribution

Ratio between chassis TS and suspension RS

2016

622.21

43.48:56.52

1.59

2017

1185.59

42.98:57.02

3.02

2018

1629.64

42.88:57.12

4.15

IDEAL

X

42.64:57.36

X

Determination, Validation, and Dynamic Analysis of an Off-Road …

59

This criterion should be used in conjunction with the increase of the mass for adding new lockings. Changes in the regulation of the Baja SAE competition in 2017 implied a larger amount of tubes in the chassis which made the mass increase inevitable. The stiffness gain between 2016 and 2017 years was beneficial since there was a mass increase of 23.4% and the error in the lateral load distribution became 1%. However for the 2017 and 2018 years, there was a mass increase of 10%, but the error of the lateral load distribution staying in the range of 0.7%. Thus for this suspension set, a chassis with torsional stiffness about 3 times the suspension roll stiffness provides the best relationship between minimization of the error of lateral load distribution and mass gain. Still it appears in few works addressing the relationship of chassis torsional stiffness and suspension roll stiffness, but it is a consensus that each car has its ideal relationship. More recent studies have concluded that there is a direct proportion between these quantities and not between the difference of suspension roll stiffness and chassis torsional stiffness, as discussed in [8]. For the suspension used between 2016 and 2018 years by TEC Ilha Baja SAE team, a chassis with torsional stiffness 3 times greater than the suspension roll stiffness would make the distribution of lateral load very close to ideal one without too much mass increasing.

4 Conclusions The calculation of the torsional stiffness is an important vehicle design parameter, and the purpose of this test is to validate the finite element model used in the design of the TEC Ilha Baja SAE team chassis. In addition, the data processing still allows the evaluation of the percentage of roll stiffness in relation to the lateral load transfer of the vehicle. The development of a mathematical model to assess how much torsional stiffness influences lateral load distribution can be an alternative design methodology. The comparison between experimental and numerical data was satisfactory, within the indicated in the literature, which proves the validity of the idealized test. The development of a mathematical model to evaluate how the torsional stiffness influences the lateral load distribution is another tool to evaluate the impact of adding new lockings for the vehicle performance with regard to the lateral dynamics and mass. This is an initial model, containing a number of simplifications, so its development needs to be continued. Acknowledgements For the partnership UNESP—São Paulo State University “Júlio de Mesquita Filho”—Santander Bank, and Ilha Solteira Engineering Faculty for material and financial support and also the TEC Ilha Baja SAE team for technical support and collaboration. Responsibility Notice The authors are the only responsible for the material included in this paper.

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References 1. Filho, Avelino A.: Elementos finitos a base da tecnologia CAE Análise Dinâmica. Ética, São Paulo (2005) 2. Moraes, M.C., Souza, A.C.G.F., Menezes, M.A.: Determinação da rigidez torcional de um veículo off road por métodos experimentais e computacionais. XXV Congresso Nacional de Estudantes de Engenharia Mecânica, Brasília, Distrito Federal (2018) 3. Thompson, L.L., Lampert, J.K., Law, E.H.: Design of a twist fixture to measure the torsional stiffness of a Winston cup chassis. SAE Technical Paper 1998-983054 (1998). https://doi.org/ 10.4271/983054 4. Sampò, E.: Modelling chassis flexibility in vehicle dynamics simulation. Ph.D. thesis (1991) 5. Moraes, M.C., Menezes, M.A.: Rigidez torcional de um veículo BAJA SAE por métodos numéricos e experimentais. Congresso SAE Brasil 2017, São Paulo (2017) 6. Costa, J.A.: Estudo da rigidez torcional do quadro de um formula SAE por análise de elementos finites (2012) 7. Milliken, D.L., Milliken, W.F.: Race car vehicle dynamics (1995) 8. Danielsson, O., Cocanã, A.G.: Influence of body stiffness on vehicle dynamics characteristics in passenger cars. 2015. 82 f. Dissertation (Master degree). Automotive Engineering, Chalmers University of Technology, Göteborg (2015) 9. SUSPENSION, All About. Suspension explained. Available in: https://suspensionexplained. blogspot.com/p/suspension-basics.html. Access in: 26 June 2019 10. Moraes, M.C., Souza, A.C.G.F., Menezes, M.A.: Structural dimensioning of an off-road vehicle suspension through drop test. 25th ABCM International Congress of Mechanical Engineering (to be presented), Uberlândia, Minas Gerais 11. Deakin, A., Crolla, D., Ramirez, J.P., Hanley, R.: The Effect of chassis stiffness on race car handling balance. SAE Technical Paper 2000-01-3554 (2000). https://doi.org/10.4271/200001-3554

Experimental Investigation of the Dynamics of a Ropeway Passing Over a Support Siegfried Ladurner, Markus Wenin, Daniel Reiterer, Maria Letizia Bertotti and Giovanni Modanese

Abstract The dynamics of ropeway vehicles of an aerial ropeway is investigated experimentally by a set of measurements including velocities of the hauling cable at the driving disk and the running gear and time-dependent deviation angles of the hangers. A measurement of the damping characteristics of vehicle oscillations shows that both damper and friction resistance at the bolt play a role regarding the dynamics of a ropeway vehicle passing a support. The results allow the development of accurate computational tools for future simulation tasks. Keywords Cable railway · Experimental investigation of vehicle oscillations

1 Introduction The motion of a ropeway vehicle can be viewed as a many-body problem in classical mechanics, characterized in particular by the presence of many mechanical constraints (deterministic part of the problem) and random effects like wind or S. Ladurner · D. Reiterer Doppelmayr Italia, Industriezone 14, 39011 Lana BZ, Italy e-mail: [email protected] D. Reiterer e-mail: [email protected] M. Wenin (B) CPE Computational Physics and Engineering, Weingartnerstrasse 28, 39011 Lana BZ, Italy e-mail: [email protected] M. L. Bertotti · G. Modanese Faculty of Science and Technology, Free University of Bozen/Bolzano Piazza Università 1, 39100 Bolzano BZ, Italy e-mail: [email protected] G. Modanese e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_6

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temperature fluctuations (stochastic part of the problem, [4, 5]). A detailed description of the dynamics is therefore a complicated task, and simplifications of the model are needed to obtain in a reasonable time results of practical relevance (as required by the standards [1, 2]). This implies the use of parameters in the equations, which must be determined by experiments/measurements. In this paper, we present results obtained by measurement at the already existing plant in Voeran (South Tyrol, Italy). We show and discuss the results of the damping behavior of a cable railway vehicle, the measurement of the velocity, and the deviation angle of the vehicle passing over the support. These results are required when one gives up the quasi-static approach to set up a more realistic description of the ropeway dynamics or to optimize the system parameters [3, 6, 7].

2 Stationary Vehicle: Oscillations and Damping We consider the damped oscillations of the ropeway vehicle by a measurement of the deviation as a function of time for both the empty and loaded vehicles, respectively. The running gear (physical pendulum) was blocked by the support cable brake, the vehicle was deviated from the vertical (φ(0)), and the oscillations were shot. In a standard theoretical approach, there are a few parameters to describe the damped oscillations of a physical pendulum, in particular the reduced pendulum length l p and parameters related to the friction resistance. For a damper with constant friction moment Md and a dynamic friction resistance of the hanger bolt μ, the differential equation for the deviation angle φ reads ¨ + g sin(φ(t)) + l p φ(t)



 Md ˙ + μg sign(φ(t)) = 0. m

(1)

The parameters Md and μ can be extracted conveniently and accurately from measurements of vehicle oscillations of a stationary vehicle (here g denotes the earth acceleration and m denotes the mass of the pendulum. The sign function ensures the dissipative character of the friction resistance). The reduced pendulum length l p depends weakly on the mass, which is also derivable from the φ(t) curves in Fig. 1 (one obtains l p for the empty/loaded vehicle 6.3/6.6 m). The latter shows, in addition to the measured data points, the results of a numerical integration of Eq. 1 with appropriate parameters. Except for small angles at the last oscillation period, there is a good agreement between measured angles and simulation. This discrepancy results from our simplified friction model which does not capture the difficult problem of the transition from dynamic to static friction. On the other hand, the experiments have shown that the aerodynamic drag can be neglected.

Experimental Investigation of the Dynamics of a Ropeway Passing Over a Support

63

empty (1970 kg) loaded (1900 kg + 2070 kg charge)

10

5

0

-5

-10

-15 0

5

10

15

20

t [s]

Fig. 1 Measurement of the damped oscillations of the stationary vehicle 1 for two different loadings. The dots indicate the measured values, and the dashed lines indicate the simulation results

3 Measurement of the Velocity and the Deviation Angle of the Moving Vehicle 3.1 Velocity of the Hauling Cable The velocity of the hauling cable at the driving station (mountain) was measured directly and compared with the theoretical default velocity (see Fig. 2). The hauling cable is spanned with a damped counterweight placed in the valley station. One can see from Fig. 2 that in the acceleration/deceleration phases, the hauling cable begins to oscillate, leading to small deviations from the default velocity. Furthermore, we measured the velocities of the vehicles itself with the setup shown in Fig. 3. The results are given in Fig. 4 for the ascent and Fig. 5 for the descent case and are interesting for several reasons. First, one can see by comparison with the theoretical velocity (the same as at the driving disk) that the vehicle accelerates at the support. This is due to a change of the hauling cable tension and is experimentally reproducible as well as theoretically understandable. Apart from this, however, there are small amplitudedamped oscillations of stochastic nature, in which different damping channels (air drag, hauling cable suspensions, counterweight) play an important role.

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v [m/s]

8

6 0.2

a [m/ ]

4

0.0 -0.2 -0.4

2 -0.6 900

1000

1100

1200

1300

s [m] 0

900

1000

1100

1200

1300

1400

s [m]

Fig. 2 Measurement of the velocity of the hauling cable at the driving disk in the mountain station (black dots). The dashed line gives the theoretical default velocity profile with piecewise constant accelerations/decelerations, as shown in the inset. The latter shows clearly that the hauling cable begins to vibrate when the driving velocity changes (the blue line is derived from the measured velocity). The colored bars show the support head position for the descent (light violet) and ascent (light red)

3.2 Deviation Angles The deviation angles φ were measured when the vehicles pass the support. The scene was recorded with a film camera and the data evaluated thereafter. To obtain correct angles free of distortion, we needed reference directions, for which we used the free-hanging vehicles at different positions in space (see Fig. 8). The results of these measurements are given in Figs. 6 and 7, respectively, where a positive angle always means deviation in the velocity direction. Fig. 6 shows the deviation angles for both vehicles in the descent case. The amplitude is smaller for the descent as for the ascent case. An explanation of the measured quantities must take into account several non-trivial effects: (i) the position-dependent velocity of the running gear (including perturbations from hauling cable oscillations), (ii) the curvature of the trajectory, and (iii) the damping and the rotation in space of the running gear, which also distress the hanger and the vehicle. As a consequence, the damper can act for the vehicle as an energy source instead of an energy sink, when it passes over ˙ ˙ − ψ(t)], ˙ the support (one has to replace in Eq. 1, sign(φ(t)) → sign[φ(t) where ˙ ψ(t) is the angular velocity of the running gear). We also carried out a numerical

Experimental Investigation of the Dynamics of a Ropeway Passing Over a Support

65

Fig. 3 Measurement of the vehicle velocity (veh. 1): a shows the running gear where a magnetic sensor (red arrow) is installed to register the rotations of a track roller. b To reduce the errors in the vehicle localization, we performed a position coincidence measurement relative to the support head. c The velocity versus hauling cable length was recorded by a video camera inside the vehicle

integration of the equation of motion for a damped pendulum with time-dependent suspension point position, which confirms the experimental findings sufficiently well. The derivation of the equation of motion can be found in [6], where the damping in Eq. 1 is correctly generalized for the running vehicle. Deviations between theoretical predictions and measurements are mainly caused by the inexact treatment of the hauling cable dynamics. In theory, the deviation angles φ for one driving direction should be equal for both vehicles. As Figs. 6 and 7 show, there are differences between the different vehicles with a maximum of about 3◦ caused by probably unwanted asymmetries in the damper strength, etc.

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theor. velocity meas. 1 (empty)

8

meas. 2 (loaded) meas. driving disk

v [m/s]

7

6

5

4

-100

-50

0

50

100

s [m]

Fig. 4 Measurement of the velocity of the running gear during the ascent. s = 0 corresponds to the center of the support head 9 theor. velocity meas. 1 (empty)

8

meas. 2 (empty)

v [m/s]

7

6

5

4

-50

0

50

100

150

s [m] Fig. 5 Measurement of the velocity of the running gear during the descent of the vehicle. The peak at the support position is caused by the change in the hauling cable tension, when the vehicle passes over the support

Experimental Investigation of the Dynamics of a Ropeway Passing Over a Support

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10 simul. (theor. v) simul. (meas. v)

5

meas. (veh. 1) meas. (veh. 2)

0

-5

-10

-60

-40

-20

0

20

40

60

s [m] Fig. 6 Measurement of the deviation angle for the descent. The measurement errors were estimated trough the spatial resolution of the video film. Two numerical simulations are shown: in the first (violet dashed line), we assume the same velocity of the running gear as at the driving disk. The deviation angle in front of the support comes from a constant deceleration. In the second simulation (black dashed line), we use a measured velocity profile, however, it remains the problem of unknown initial conditions of the vehicle (pendulum)

4 Conclusion In this work, we presented the results of an experimental investigation of an areal cable railway vehicle passing over a support. We made and discussed several measurement arrangements of the vehicle velocities and oscillations. We clarified the influence of the mass on the oscillations—damping of the vehicles. A simple numerical simulation of the vehicle dynamics (oscillations) matches sufficiently well with the measured data and helps to understand the physics of the damped pendulum whose suspension point moves along with a certain trajectory (essentially given by the support head geometry) with a position-dependent velocity. The obtained results are helpful for the development of accurate simulation tools.

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10

meas. (veh. 1) meas. (veh. 2)

5

0

-5

-10

-15 -60

-40

-20

0

20

40

60

80

s [m] Fig. 7 Deviation angle for the ascent. In this case, the amplitude is greater as for the descent. The simulation does not contain the small oscillations in front of the support, because we assume a constant velocity and a straight line for the trajectory of the running gear

A B

C

veh. 1

veh. 2 D

Fig. 8 Left picture: four positions in space where the vehicles were stopped and photographed to define a vertical direction (the free-hanging vehicle) needed for the deviation angle measurement. Due to the small oscillations of the system (two support cables for each vehicle and the hauling cable), the static configurations have shown still max. fluctuations of ca. ± 0.3◦ . The support head has a length of 23.7 m, the tangent angles of the vehicle trajectory are α = 6◦ (mountain side, left in the picture) and α = 44◦ at the valley side (right in the picture). The distances from the support position to the stations are about 1 km. Right picture: sequence of positions with the evaluation of deviation angles for the descent of vehicle 1

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Acknowledgements S. Ladurner, D. Reiterer, and M. Wenin acknowledge financial support by the Amt für Forschung und Innovation der Provinz Bozen, Italy (this work is a part of the project “Steigerung der Geschwindigkeit und des Fahrkomforts bei der Stützenüberfahrt von Seilbahnanlagen”).

References 1. Bryja, D., Knawa, M.: Computational model of an inclined aerial ropeway and numerical method for analyzing nonlinear cable-car interaction. Comput. Struct. 98, 1895–1905 (2011) 2. CEN-Norm: Sicherheitsanforderungen für Seilbahnen für den Personenverkehr. Amtsblatt der EU C51 (2009) 3. Czitary, E.: Seilschwebebahnen (2. Auflage). Springer Verlag, Wien (1962) 4. Kopanakis, G.: Schwingungen bei Seilbahnen. Internationale Seilbahn-Rundschau 1, 22 (2010) 5. Volmer, M.: Stochastische Schwingungen an ausgedehnten Seilfeldern und ihre Anwendung zur Spurweitenberechnung an Seilbahnen. Dissertation ETH Nr. 13379 (1999) 6. Wenin, M., Windisch, A., Ladurner, S., Bertotti, M.L., Modanese, G.: Optimal velocity profile for a cable car passing over a support. Eur. J. Mech. A. Solids 73, 366–372 (2019) 7. Wenin, M., Windisch, A., Ladurner, S., Bertotti, M.L., Modanese, G.: Optimization of the head geometry for a cable car passing over a support. In: Engineering Design Applications II. Springer International Publishing (2020)

Environmental Fatigue Analysis of the Feedwater Piping System of a BWR-5 Laura Guadalupe Carbajal-Figueroa, Salatiel Pérez-Montejo, Alejandra Armenta-Molina, Gilberto Soto-Mendoza, Luis Héctor Hernández-Gómez, Juan Alfonso Beltrán-Fernández and Pablo Ruiz-López Abstract The fatigue analyses are usually carried out with experimental data obtained in dry conditions. Nevertheless, a wet environment significantly affects the fatigue life of the elements exposed to these conditions. In this paper, the structural integrity of a feedwater piping system of a boiling water reactor (BWR-5), under wet environment fatigue conditions during 40 and 60 years of operation, was evaluated. A structural analysis of the piping system, between the feedwater nozzle and the adjacent piping support to the reactor pressure vessel, was done. The transient conditions, which are developed during the start-up and shut down, were taken into account. The fatigue analysis was developed in accordance with the Section III of the corresponding ASME Code. The environmental factors, which are related with the contact between the water and the internal surfaces of the piping system, were considered as penalty parameters. The procedure explained in the NUREG 6909 was followed. The results showed that the environmental conditions increased the cumulative usage factor, in comparison with those calculated in dry conditions. Keywords Piping system fatigue · Piping supports · Piping flexibility · Numerical analysis · Finite element method

L. G. Carbajal-Figueroa · S. Pérez-Montejo · A. Armenta-Molina · G. Soto-Mendoza · L. H. Hernández-Gómez (B) · J. A. Beltrán-Fernández Instituto Politécnico Nacional, Sección de Estudios de Posgrado e Investigación, Escuela Superior de Ingeniería Mecánica y Eléctrica Zacatenco, Unidad Profesional Adolfo López Mateos, Edificio 5, Segundo Piso, Colonia Lindavista, Alcaldía Gustavo A. Madero, Ciudad de México 07738, México A. Armenta-Molina e-mail: [email protected] P. Ruiz-López Comisión Nacional de Seguridad Nuclear y Salvaguardias, Dr. José Ma. Barragán no. 779, Colonia Narvarte, Alcaldía Benito Juárez, Ciudad de México 03020, México e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_7

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Abbreviations ASME BWR CUF CUFen F en LWR NPP NUREG PWR RPV S alt

American Society of Mechanical Engineers Boiling Water Reactor Cumulative Usage Factor Environmental Cumulative Usage Factor Environmental Factor Light-Water Reactor Nuclear Power Plant Nuclear Regulatory Commission Pressurized Water Reactor Reactor Pressure Vessel Alternate Stress Intensity

1 Introduction In the industrial processes, cyclic loads, on mechanical components and structures, are generated by the transient conditions that take place during the diverse conditions of operation [1]. The nuclear industry is not an exception. Many of the components that are involved in the process of power generation in a nuclear power plant (NPP) are working in an aqueous environment [2]. Under these conditions, the fatigue life is reduced. This is the case of the piping components and the non-piping passive components like the reactor pressure vessel, its internals and the feedwater nozzles. They are under cyclic loads and in contact with water [3]. In the case of this paper, the feedwater line class 1 piping was selected. The main objective was the evaluation of the accumulated damage after 40 and 60 years of operation [4]. In accordance with the Nuclear Regulatory Commission, NUREG 6909 [5], the materials used in the light-water reactor (LWR) reactors are affected by different conditions of operation (normal, emergency and extraordinary). In the American Society of Mechanical Engineer (ASME) Boiler and Pressure Vessel Code Section III, fatigue design curves are based on strain-controlled fatigue tests in dry conditions. For this reason, they are limited and may not be useful for NPP aqueous environment. This is the case of those components which are under fatigue loading and in contact with water. Even now, there are data obtained in the U S and Japan, and it has been demonstrated that the aqueous environment in the LWR can result in a reduction of the fatigue life in the materials [5]. Nowadays, research related with environmental fatigue has been developed. Mehta and coworker [6] have determined an environmental correction factor. Besides, a fatigue evaluation methodology was proposed with different applications, like piping and nozzles in LWR reactors. The ASME Section III NB-3600 and NB-3200 considered statistical models that calculate the probability of fatigue failures in five pressurized water reactor (PWR) and two boiling water reactor (BWR) plants after

Environmental Fatigue Analysis of the Feedwater Piping …

73

40 and 60 years of operation. Besides, a probabilistic fracture mechanics code was developed to simulate the initiation and the growth of fatigue cracks [7]. Regarding experimental work, SCKβ·CEN has developed environmental fatigue testing in simulated PWR primary water in the framework of INCEFA-PLUS. Its purpose is the increment of safety levels in nuclear power plants by covering gaps in environmental fatigue assessment [8].

2 Statement of the Problem The structural integrity of a feedwater piping system of a BWR-5, under environmental fatigue conditions, was evaluated. For this purpose, the fatigue loads that take place during the start-up and shutdown were considered. In this case, the transient stress fields that take place were evaluated. Such evaluation was focused on the internal areas of the piping system, because they are in contact with water. The environmental cumulative usage factors (CUFen ) for 40 and 60 years were estimated.

3 Finite Element Model There are four feedwater nozzles which are located symmetrically at a BWR-5 reactor pressure vessel. In order to consider its thermal expansion, a quarter model was developed (Fig. 1) with ANSYS® Space Claim. In this way, it was taken into account the thermal displacement of the nozzle. This model was complemented with the adjacent feedwater piping system. Its nominal diameter and schedule were 12 and 80, respectively. The free extreme of the pipe was fixed. The materials for each component of the model are shown in Fig. 2, and their material properties were assigned in accordance with the ASME Section II [9]. A finite element model was developed. It has 41,585 elements and 94,910 nodes. The average of the mesh quality is 0.78. The SOLID87, SOLID90 and SURF152 elements were used in the transient thermal analysis. Besides, the SOLID187 and SOLID186 elements were used in the structural analysis [10].

4 Transient Loading Cases The refuelling of the reactor, that takes place during each outage, was considered. Therefore, a start-up and a shutdown were considered. They are illustrated in Figs. 3 and 4, respectively. For this purpose, 27 cycles and 40 cycles have been estimated to take place during 40 years and 60 years of operation, respectively. There is a temperature drop after 18,000 s (Fig. 3). This caused severe thermal stresses. Regarding Fig. 4, the variation of the pressure and temperature is smooth.

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Feedwater nozzle

Safe end

Elbow 2 Elbow 1 Pipe Support

Elbow 3 Fig. 1 Isometric of the feedwater piping system

Safe end SB-366 Pipe SA-376

Thermal sleeve A182 F304

Vessel and nozzle SA-508

Fig. 2 Materials of the feedwater piping system

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Fig. 3 Start-up transient of a BWR [11]

Fig. 4 Shutdown transient of a BWR [11]

5 Results According to the numerical analysis, the piping system which was analysed has four points of interest. Three of them are the internal surfaces of the elbows; the last one is the safe end. The maximum stresses were developed in the internal surface of the elbows and in the safe end. However, the first elbow, which is closed to the reactor

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nozzle, is the zone with the greatest peak stress. This is illustrated in Fig. 5b. Table 1 summarizes all the results.

Fig. 5 Stress intensity field, a in the safe end and b in the elbows

Table 1 Peak stresses at the points of interest in the feedwater line –

Safe end (MPa)

Elbow 1 (MPa)

Elbow 2 (MPa)

Elbow 3 (MPa)

Start-up

77.14°

198.2°

181.16°

112.01°

Shutdown

77.45°

201.48°

184.3°

114.82°

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It is important to observe that these stressed areas are in contact with water (Fig. 5). Accordingly, fatigue environmental conditions were developed. The critical situation took place in the first elbow during the shutdown. For this reason, the cumulative usage factor (CUF) and environmental cumulative usage factor (CUFen ) were evaluated at this point.

6 Evaluation of the Environmental Cumulative Usage Factor The cumulative usage factor was determined with the numerical results for 40 and 60 years of operation. The CUF was obtained in accordance with the Miner’s rule and the Section III of the ASME code. In this case, two transient events take place, the start-up and shutdown. In each event, the same number of load cycles take place [12]. Then, the CUF of each event was estimated, and they were added in accordance with the following relationship. CUF =

M  ni Ni i=1

(1)

In accordance with the Subsection NB-3216.1 of the Section III of the ASME code, the alternate stress intensity (S alt ) was obtained with the following equation. Salti j = 0.5 Sri j

(2)

The table of design fatigue curves of the ASME Appendix I Section III was used. The fatigue life was determined with this data [13]. The results are shown in the following table. Table 2 Cumulative usage factor for 40 and 60 years in the first elbow

–

Start-up

Shutdown

Range stress difference (S r )

198.2° MPa

201.48° MPa

Alternating stress intensity (S alt )

99.1° MPa

100.74° MPa

n number of cycles after 40 years

27 cycles

27 cycles

n number of cycles after 60 years

40 cycles

40 cycles

N fatigue life

9.77E6 cycles

6.67E6 cycles

CUF after 40 years of operation

2.76E-6

4.05E-6

CUF after 60 years of operation

4.1E-6

5.99E-6

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Table 3 Environmental CUF for 40 and 60 years at the first elbow

–

CUF

F en

CUFen

40 years of operation 60 years of operation

6.86E-6

1.6273

1.116E-5

1.008E-5

1.6273

1.64E-5

The environmental CUF was obtained in accordance with the NUREG 6909 [5]. The most critical point is localized at the first elbow. The CUFen was evaluated during the start-up. The environmental factor (F en ) was calculated with the following equation:   Fen = exp −T ∗ ε·∗ O ∗

(3)

This relationship has been proposed in the NUREG 6909 [14]. It was considered that the material of the piping system was made with SA-376 stainless steel [11]. The parameters for this case were the following: T ∗ = 0.356 O ∗ = −9.7699 ε·∗ = 0.14 As a result, the environmental factor was: Fen = 1.6273 The F en for the shutdown transient was determined in the same way. In this case, it was: Fen = 1.6273 Finally, the environmental cumulative usage factor was calculated with the following equation: CUFen = U1 ∗ Fen,1 + U2 ∗ Fen,2 + · · · + Un ∗ Fen,n

(4)

Table 3 shows the CUFen for 40 and 60 years of operation.

7 Conclusions The elbows provide flexibility to the piping system. In this way, the level of the resultant stresses is reduced. However, there is stress concentration at such elbows. For this reason, this analysis was focused at these points. The results showed the CUFen is 62.7% bigger than the CUF. However, it is lower than 1 in all the evaluations which were carried out. Therefore, the feedwater piping system can operate during 60 years. It has to keep in mind that in the evaluation, the

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start-up and shutdown transients were considered. However, there are other transient events, which have to be considered. Even though the CUFen is not significative, preventive actions must be taken in order to avoid any fatigue damage before a leak or crack. The examination and the inspection according to the ASME Code Section XI should be considered in risk areas. Statement The conclusions and opinions stated in this paper do not represent the position of the National Commission on Nuclear Safety and Safeguards, where the co-author P. Ruiz-López is working as an employee. Although special care has been taken to maintain the accuracy of the information and results, all the authors do not assume any responsibility on the consequences of its use. The use of particular mentions of countries, territories, companies, associations, products or methodologies does not imply any judgment or promotion by all the authors. Acknowledgements The grant for the development of the project 211704 awarded by the Consejo Nacional de Ciencia y Tecnología (CONACYT) is kindly acknowledged.

References 1. U. S. Nuclear Regulatory Commission: NUREG/CR-6815 Review of the Margins for ASME Code Fatigue Design Curve Effects of Surface Roughness and Material Variability. Washington D.C. (2003) 2. Féron, D., Olive, J.M. (Eds.): Corrosion issues in light water reactors: stress corrosion cracking, vol. 51. Elsevier (2007) 3. NEA Component Operational Experience, Degradation and Ageing Programme (CODAP): Second Term (2015–2017) Status Report (2019) http://www.oecd.org/officialdocuments/ publicdisplaydocumentpdf/?cote=NEA/CSNI/R(2019)7&docLanguage=En. Accessed 15 May 2019 4. U. S. Nuclear Regulatory Commission: NUREG/CR-6674 Fatigue Analysis of Components for 60-Year Plant Life Washington D.C. (2000) 5. U. S. Nuclear Regulatory Commission: NUREGCR-6909 Rev. 1. Effect of LWR water environments on the fatigue life of reactor materials. Washington D.C. (2018) 6. Mehta, H., Gosselin, S.: Environmental factor approach to account for water effects in pressure vessel and piping fatigue evaluations. Nuc. Eng. Des. 181(1–3), 175–197 (1999). https://doi. org/10.1016/s0029-5493(97)00344-0 7. Simonen, F., Khaleel, M., Phan, H., Harris, D., Dedhia, D., Kalinousky, D., Shaukat, S.: Evaluation of environmental effects on fatigue life of piping. Nuc. Eng. Des. 208(2) (2001). https:// doi.org/10.1016/s0029-5493(01)00373-9 8. Vankeerberghen, M., Marmy, P., Bens, L.: PWR fatigue testing at SCK·CEN in the framework of INCEFA + . In: Abdel Wahab, M. (eds) Proceedings of the 7th International Conference on Fracture Fatigue and Wear. FFW 2018. Lecture Notes in Mechanical Engineering. Springer, Singapore (2018) 9. Boiler, A.S.M.E., Code, P.V. Sec. II: Materials. Part D: Properties (Customary). American Society of Mechanical Engineers. New York (2007)

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10. ANSYS® [Mechanical Application] 2019 R1, help system [Mechanical User’s GuideChapter 7. Element Library] ANSYS, Inc. https://ansyshelp.ansys.com/account/secured? returnurl=/Views/Secured/corp/search.html?q=SOLID87&lang=en&start=25&rows=25& pg=2&qt=”default” 11. Evon, K., Gilman, T., Walter, M.: Fatigue monitoring of BWR feedwater nozzles. In: ASME 2013 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers Digital Collection (2014). https://doi.org/10.1115/pvp2014-28140 12. Boiler, A.S.M.E., Code, P.V.: Section III Division 1-Subsection NB: Class 1 Components. Rules for construction of Nuclear power plant components, New York (2007) 13. Boiler, A.S.M.E., Code, P.V.: Section. III Division 1—appendices rules for construction of nuclear facility components. New York (2007)

Structural Vibrations in a Building of a Nuclear Power Plant Caused by an Underground Blasting Alejandra Armenta-Molina, Abraham Villanueva-García, Gilberto Soto-Mendoza, Salatiel Pérez-Montejo, Pablo Ruiz-López, Juan Alfonso Beltrán-Fernández, Luis Héctor Hernández-Gómez and Guillermo M. Urriolagoitia-Calderón Abstract In this paper, a methodology has been proposed to obtain the response spectrum of a nuclear building caused by underground detonations. For the problem at hand, a reinforced concrete building containing a boiling water reactor was postulated. It was constructed on a hard soil with a zero-period acceleration of 0.26 g. Its structural damping was 7%. The building is subjected to horizontal vibrations originated from a detonation of 15 tons of TNT, located at a distance of 2000 m from the nuclear power plant site. Initially, an analytical evaluation was carried out to obtain the horizontal vibrations. In a second step, a foundation soil model was developed with the finite element method in conjunction with an explicit approach. ANSYS® Autodyn® code was used. The response spectrum due to the underground detonation was developed with the data obtained. It was compared with the safe shutdown earthquake (SSE) response spectrum and the operating basis earthquake (OBE) response spectrum. The results show that the seismic response spectrums envelop the explosion response spectrum. A resonance did not develop. The horizontal velocity is below the limits established by the regulations considered in this paper. Under this condition, the postulated building is safe. Keywords Seismic waves · Peak particle velocity · Nuclear power plant buildings · Blast response spectra

A. Armenta-Molina · A. Villanueva-García · G. Soto-Mendoza · S. Pérez-Montejo · J. A. Beltrán-Fernández · L. H. Hernández-Gómez (B) · G. M. Urriolagoitia-Calderón Instituto Politécnico Nacional, Sección de Estudios de Posgrado e Investigación, Escuela Superior de Ingeniería Mecánica Unidad Zacatenco, Edificio 5, 2do piso. Unidad Profesional Adolfo López Mateos. Col, Lindavista, Alcaldía Gustavo A. Madero, Ciudad de México 07738, México P. Ruiz-López Comisión Nacional de Seguridad Nuclear y Salvaguardias, Dr. José Ma. Barragán No. 779, Colonia Narvarte. Alcaldía Benito Juárez, Ciudad de México 03020, México e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_8

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Abbreviations ASME BWR FEM g NPP OBE SSC U.S.NRC

American Society of Mechanical Engineers Boiling Water Reactor Finite Element Method Gravity acceleration Nuclear Power Plant Operating Basis Earthquake Safe Shutdown Earthquake United States Nuclear Regulatory Commission

1 Introduction Structures can be affected by dynamic loads due to various circumstances. Seismic loads and those generated by blasting, among others, are within this range of loads. The vibration produced by an underground blasting is mainly affected by several parameters: (1) the way in which the explosion takes place, (2) the distance between the focus of the explosion and the structure, (3) the geological properties and (4) the characteristics of explosives, mainly [1]. The vibrations caused by an explosion are mainly characterized by two important parameters: the maximum particle velocity (PPV) and frequency. Ground vibration varies with the distance to the blasting point, the depth of explosion and the magnitude of the dynamic load [2]. In the case of a controlled blasting, it is generated by sequential explosions that are separated by short periods and a specific amount of explosive is detonated. If some of these vibrations interact directly with a structure, it can vibrate under resonance conditions. Under these conditions, a collapse of the structure may occur. The regulatory guide 1.61 “Damping Values for Seismic Design of Nuclear Power Plants” of the US NRC establishes the damping levels. For the horizontal soil component of the design response spectrum, without effects of soil-structure interaction, a 7% damping is taken into account in a safe shutdown in earthquake (SSE). Alternatively, a 4% damping is taken into account for a similar perturbation in the operating basis earthquake (OBE). These two parameters are the standards considered in the evaluation of nuclear facilities [3]. There is limited information in the open literature related with the regulations for the evaluation of the structural integrity of a building that contains a BWR-5 reactor, when subsoil vibrations occur. Because of this, information related with the mining industry was considered. Some of the international standards related to this topic are the following: (1) UNE 22-381-93 (Spain) [4] and USBM-RI-8507 (USA) [5].

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Table 1 Damage prevention criterion of the standards considered in this paper [4, 5] Maximum horizontal velocity Type of structure

UNE 22-381

USBM R18507

Range of Frequency

Low frequencies

0–10 Hz Buildings used for commercial purposes, industrial buildings and buildings of similar design

I

20 mm/s

Dwellings and buildings of similar design and/or occupancy

II

9 mm/s

Structures that, because of their particular sensitivity to vibration, cannot be classified under lines I and II and are of great intrinsic value (e.g. listed buildings under preservation order)

III

4 mm/s

19 mm/s

12.5 mm/s

Table 1 comperes the standards mentioned above. Different kinds of structures, buildings and historic monuments have been considered. For the purpose of this paper, type III structures were taken into account, because they are sensible to vibration.

2 Statement of the Problem A secondary contention of a BWR-5 (Fig. 1) was considered. It was built on hard soil. The dynamic response spectrum for a NPP considers that the zero-period acceleration (ZPA) is 0.26 g [3]. Such building was loaded with horizontal vibrations generated by an underground detonation. The secondary containment building is a rectangular structure. Its height is 55 m. 10 m of the building are below ground level and the length on each side is 45 m [6]. This structure was loaded with a blast produced with 15 tons of TNT. The blast point was located 2 km far away from the reactor building and 10 m below the ground level. The ground material is basalt [6]. As the blast took place, elastic waves were generated and they propagated across the ground (Fig. 2). In accordance with the 1.61 regulatory guide, a 7% damping was considered for a safe shutdown earthquake (SSE) and 4% for an operating basis earthquake (OBE) [3].

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Refuel floor Wall thickness 1m Secondary Containment Structure Wasted fuel 55m

Reactor pessure vessel Core Primary Containment Structure Core shroud Water

10 m

Ground level

Foundation slab thickness 4 m

Fig. 1 Main characteristics of a BWR-5 secondary containment structure [6, 7]

Coordinates

Gauges

(1500000,5000)

10m

(1000000,5000)

11

8

TNT

(1500000,0)

(1000000,0)

10

7

(2000000,5000) 14 (2000000,0) 13

2 km

Distance from the detonation point to the reactor building

Reactor building

Detonation Point

Fig. 2 Schematic arrangement of the dominium of interest

3 Methodology The methodology proposed for this analysis was the following: 1. The mining regulations were followed. They establish the guidelines to evaluate the structural integrity of buildings, which are loaded with elastic waves generated by explosions. 2. A finite element model was developed with ANSYS® Autodyn® .

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3. The response spectrums were obtained with the results of the finite element analysis. For this purpose, PRISM code was used. 4. A comparison was made between the blast response spectrum and the response spectrum of a safe shutdown earthquake (SSE). A similar evaluation was done between the operating basis earthquake (OBE) and response spectrum of a blasting mentioned above.

4 Structural Modelling ANSYS® Autodyn® [8] was used. It allows to simulate dynamic loads of great magnitude, that take place in a short time. These loads can be generated by impact, large pressures or explosions. This code can model solids with Lagrangian elements. Such elements are adequate for structural analysis. For the problem at hand, they were used to model the underground with 600,000 elements. 601,901 nodes were required. On the other hand, as air is a fluid, an Euler approach was followed. In this case, the mesh is fixed and it is a regular rectilinear mesh of 600,000 elements. The air, which was simulated, flows through the grid. Figure 3 illustrates this model. (a) Soil (Basalt) is the green area. In this case, the Lagrange solver, which has been designed to solve structural response analysis, was used. (b) Air is the blue area. This dominium was evaluated with the Euler solver. It is adequate for the simulation of a wave propagation in a fluid. (c) Gauges or points of evaluation are represented by pink rhombuses. They are placed in the ground area and collect data of the parameters evaluated. It can be velocity along the x-axis and y-axis, pressure, density and temperature, mainly. (d) The TNT explosive material is represented by a semicircle. In the case of this paper, it contains 15 ton of TNT.

Boundary condiƟons [f] Air Euler [b]

TNT [d]

Gauges [c] Subsoil, Basalt Lagrange [a] DetonaƟon point [e]

Fig. 3 Dominium of analysis

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(e) The detonation point is represented by a red rhombus. (f) The boundary conditions at the top and left sides of the dominium of evaluation were considered as a flow out type. In this case, the content of the explosion leaves the environment. On the other hand, the bottom side is rigid. The reactor building is located on the right side. Therefore, a mixed condition was considered, rigid and flow out.

5 Results Figure 4 illustrates the simulation after a short period that the blast has taken place. As it can be seen, the expansion waves propagate from the detonation point in two ways. In the first, the propagation takes place through the underground. In the second, the blast wave propagates through the hole above the point of detonation. Then after, the expansion takes place, in a hemispheric shape, in the atmosphere. The response spectrum considers the displacement, the velocity, acceleration and the gravity. Therefore, the analytic evaluation requires the equation of the harmonic movement. Initially, the natural frequency was calculated with finite element model. It was 4.7 Hz. In the next step, peak velocity of the particle (PPV) was calculated  PPV = k

d 1

ω2

α (1)

Fig. 4 Wave propagation at the underground and the atmosphere a few instants after the detonation

Structural Vibrations in a Building of a Nuclear Power Plant …

 PPV = 160

2000 m

−1.6

1

15 ton 2

87

= 13.09996 mm/seg

The amplitude was estimated with following relationship u= u=

PPV 2π f

(2)

13.09996 mm/seg = 0.44360 mm 2π(4.7 Hz)

The angular frequency was ω = 2π f ω = 2π(4.7 Hz) = 29.53

(3) rad seg

In the next step, the displacement, velocity and acceleration were calculated. In this case, ϕ = 0 x = A sen(ωt) + ϕ

(4)

x˙ = A ω cos (ωt) + ϕ

(5)

x¨ = A ω2 sen(ωt) + ϕ

(6)

Ag = x(9806.61 ¨ mm/s2 )

(7)

These results were used in the PRIMS code. It is useful in the obtention of the response spectrums. The secondary containment building is a reinforced concrete building. A 4% damping was considered for the OBE response spectrum and 7% damping for an SSC response spectrum. Regarding the numerical analysis, Autodyn® calculated the displacements, velocities and accelerations through the time of evaluation at the points of evaluation. This data was introduced in the PRISM code. In this way, the explosion response spectrum was obtained. Figure 5 compares the response spectrum of the underground explosion that was obtained with ANSYS® Autodyn® with the analytical evaluation. The levels of damping considered were 4 and 7%. This is the critical damping percentage established by the normative for reinforced concrete [9]. Figure 6 compares the Operating Basis Earthquake with a 4% damping with the explosion response spectrum.

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Fig. 5 Comparison of the numerical and analytical explosion response spectrums. The natural frequency was 4.7 Hz

Fig. 6 Comparison between the explosion and OBE response spectrums, 4% damping

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Fig. 7 Comparison between the explosion and SSE response spectrums, 7% damping

Figure 7 compares the explosion response spectrum with the safe shutdown earthquake (SSE) response spectrum. A 7% damping was taken into account. In both cases, the seismic response spectrum envelopes the explosion response spectrum and resonance conditions did not take place. It can be concluded that reactor building is safe.

6 Conclusion The results of the numerical analysis showed that the horizontal velocity of the reactor building, generated by the induced vibrations, was 0.0338 mm/s. This value is lower than those established in Table 1 for type III structures. The pattern of the waves, that are generated after a detonation, is complex. Regarding the analysed case in this paper, the waves propagate through two domains, which have different mechanical and physical properties. The explicit analysis was useful, because it can be analysed the propagation of the resultant waves through the air and underground. This evaluation can be done if an Euler and Lagrange grids are coupled. The proposed methodology in this paper allows the evaluation of an underground blast at different distance and depths from structures. Levels of security can be estimated. Statement The conclusions and opinions stated in this paper do not represent the position of the National Commission on Nuclear Safety and Safeguards, where the co-author P. Ruiz-López is working as an employee. Although special care has been taken to maintain the accuracy of the information and results, all the authors do not assume

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any responsibility on the consequences of its use. The use of particular mentions of countries, territories, companies, associations, products or methodologies does not imply any judgment or promotion by all the authors. Acknowledgements The authors kindly acknowledge the grant for the development of the Project 211704. It was awarded by the National Council of Science and Technology (CONACyT).

References 1. Chopra, A.K.: Dinamics of Structures Theory and Applications to Earthquake Engineering. William J, Hall, New Jersey (1995) 2. Tripathy, G.R., et al.: Safety of engineered structures against blast vibrations: a case study. J. Rock Mech. Geotech. Eng. (2016). https://doi.org/10.1016/j.jrmge.2015.10.007 3. GUIDE Regulatory 1.61: Damping Values for Seismic Design of Nuclear Power Plants, US Atomic Energy Commission, US (1973) 4. Asociación Española de Normalización y Certificación (AENOR): Norm Spain Control of vibrations made by blastings UNE 22-381-93, AENOR, Madrid (1993) 5. Siskind, D.E., et al.: Structure Response and Damage Produced by Ground Vibration from Surface Mine Blasting. Bureau of mines, United States (1983) 6. Ruiz López, P.: Integridad Estructural de la Envolvente del Núcleo de un Reactor BWR con Defectos Distribuidos Irregularmente Tesis Doctorado Instituto Politécnico Nacional ESIMEZacatenco, México (2017) 7. López Grijalba, Y.: Análisis de fatiga asistida ambientalmente aplicado al faldón de un reactor nuclear BWR 5 Tesis Doctoral IPN ESIME-Zacatenco, México (2018) 8. ANSYS® , Lecture 1 Introduction to AUTODYN® © 2015 ANSYS Inc. Introduction to ANSYS Autodyn® part II. ANSYS Inc, ANSYS® , México 9. USNRC 10 CFR 100 (2017) U.S. NUCLEAR REGULATORY COMMISSION, § 100.23 Geologic and seismic siting criteria. https://www.nrc.gov/reading-rm/doc-collections/cfr/part100/ part100-0023.html. Accessed 29 June 2019

Analysis of an Aircraft Impact on a Dry Storage Cask of Spent Nuclear Fuel Edgar Hernández-Palafox, Juan Cruz-Castro, Yunuén López-Grijalba, Luis Héctor Hernández-Gómez, Guillermo Manuel Urriolagoitia-Calderón and Laura Guadalupe Carbajal-Figueroa

Abstract Nowadays, nuclear power plants can use a dry storage cask (DSC) for the spent nuclear fuel (NSF) on site. It is a reinforced concrete vessel, which has an internal stainless steel cylinder. Its main function is to provide a barrier against radiation, to cool down the spent fuel and to avoid nuclear fission through the internal metallic surface that contains the radioactive material. This vessel has been designed to withstand different conditions such as free fall, penetration, crushing and extreme temperatures. However, there is limited knowledge about the assessment of an aircraft impact on such vessels in the open literature. In this paper, the case of an oblique impact (45° impact angle with respect to the horizontal) of a light aircraft against a dry storage cask was considered. Its structural integrity was evaluated. The containers were considered to be located in an outdoor area. The evaluation was carried out with the commercial ANSYS® code. Keywords Spent fuel storage pool · Independent spent fuel storage installations · Aircraft impact · Structural integrity · Explicit dynamics analysis

Nomenclature CAD CATIA DSC EPRI

Computational Assisted Design Computer-Aided Three-dimensional Interactive Application Dry Storage Cask Electric Power Research Institute

E. Hernández-Palafox · J. Cruz-Castro · L. H. Hernández-Gómez (B) · G. M. Urriolagoitia-Calderón · L. G. Carbajal-Figueroa Instituto Politécnico Nacional, Sección de Estudios de Posgrado e Investigación, Escuela Superior de Ingeniería Mecánica y Eléctrica, Unidad Zacatenco, “Unidad Profesional Adolfo López Mateos”, Edificio 5, Segundo Piso, Colonia Lindavista, Alcaldía Gustavo A. Madero, 07738 Ciudad de México, Mexico Y. López-Grijalba Instituto Politécnico Nacional. Unidad Profesional Interdisciplinaria de Ingeniería, Campus Hidalgo, Carretera Pachuca-Actopan km 1 + 500, 42162 San Agustín Tlaxiaca, Hidalgo, Mexico © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_9

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FEA FEM ISFSI CSN NPP SFSP SNF USNRC δ(t) [C] F1 (t) [K] [M] MTU {R(t)} {x} ˙ {x} ¨

E. Hernández-Palafox et al.

Finite Element Analysis Finite Element Method Independent Spent Fuel Storage Installations Nuclear Safety Council Nuclear Power Plant Spent Fuel Storage Pool Spent Nuclear Fuel United States Nuclear Regulatory Commission Green function Damping matrix Time-dependent load including impact/explosion Stiffness matrix Structural mass matrix Metric Ton of Uranium Residual time-dependent load vector Structural velocity matrix Structural acceleration matrix

1 Introduction The nuclear fuel assemblies are placed inside the nuclear power plant (NPP) reactors, where they become radioactive due to the generation of fission products. The US Nuclear Regulatory Commission (USNRC) [1] mentions that upon completion of the required time, the spent nuclear fuel (SNF) assemblies have to be removed from the reactors and deposited in temporary spent fuel storage pools (SFSP), to be cooled and reduce the heat that releases radioactive decay (Fig. 1). The SNF assemblies have to be stored in the cooling pools for approximately five years, before they can be reprocessed. The USNRC [1] points out that because commercial reprocessing has never been successful in the USA, and the pools in the storage capacity of the NPPs have been exceeded, an alternative has to be followed. The Electric Power Research Institute (EPRI) [2] has reported that the nuclear energy industry has generated new independent spent fuel storage installations (ISFSI). As a result, the NPPs use the dry storage cask (DSC) for SNF on site. EPRI [2] has mentioned that at the end of the year 2011, approximately 17,300 MTU of SNF were storage into more than 1,500 DSCs. EPRI projects that there will be 32,000 MTU of SNF for the year 2020 and will be stored in approximately 2,900 DSCs. By 2060, the total of the nuclear power plants of the U.S.A. will reach the end of their renewed operating licenses. In consequence, 136,600 MTU of SNF have to be stored on site. Due to the need to continue operating under renewed licenses for 20 more years, the nuclear industry solved the problem of handling and storage of spent nuclear fuel. This situation gave rise to the design of DSC.

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Containment Structure

Polar Crane

Upper Pool Spent Nuclear Fuel

Nuclear Fuel Assemblies

BWR

Spent Fuel Storage Pool

Transfer Canal

Spent Nuclear Fuel

Fig. 1 Nuclear fuel assemblies and nuclear spent fuel in a NPP type BWR

The USNRC [3] established that the DSCs have been designed to contain radiation, reduce heat and prevent nuclear fission. They have been manufactured with an external reinforced concrete shell. It provides radiation protection. The DSC has an internal stainless steel cylinder, which is sealed (Fig. 2). Regarding the structural integrity of these vessels, it has been mentioned in [4, 5] that they have been designed against earthquakes, tornadoes, free fall, penetration, extreme temperatures and immersion. Also, Shah [6] and Hanifehzadeh et al. [7] have tested containers, producing a rollover event at an angular impact velocity during a seismic event. Nowadays, different works have been carried out on the subject of missile and aircraft impact. Siefert et al. [8] carried out a model of an aircraft to analyse the impact on the wall of a building in NPP. On the other hand, Bangash [9] described the impact of an aircraft on nuclear structures. They calculated displacements and frequencies. However, to date, there is not enough information in the open literature, about the structural integrity of the DSCs of SNF when an aircraft has impacted them. In this paper, an oblique impact of a light airplane on a DSC of SNF has been analysed. A 45° impact angle with respect to the horizontal was considered, in order to evaluate the structural integrity in the DSC.

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Stainless steel cylinder

Reinforced concrete container

Rebar Spent fuel assemblies

Fig. 2 A schematic arrangement of a dry storage cask of spent nuclear fuel

2 Dynamic Nonlinear Analysis Bangash [9] mentioned that the impact problems are governed by the dynamic coupled equations to evaluate the response of the structure, using the time increment δ(t) (Green function). In this way, the impact analysis can be described by Eq. (1). ˙ + [kin ]{δ(t)} = {R(t)} + {F1 (t)} [M]{x(t)} ¨ + [Cin ]{x(t)}

(1)

where {x} ˙ and {x} ¨ are the structural matrix of velocity and acceleration, respectively; [M] is the structural mass matrix; [C] is the damping matrix; [K] is the stiffness matrix, F1 (t) is the time-dependent load including impact/explosion load; {R(t)} = {δ Pn (t)} is the residual time-dependent load vector. Lastly, “in” denotes initial effects by the iteration.

3 Explicit Dynamics ANSYS® Explicit Dynamics [10] simulates the response of structures under dynamic loadings. It solves problems involving short-duration and severe loading, large material deformation and material failure. The explicit solution method can handle geometries with complex nonlinear contact. ANSYS® Explicit Dynamics uses explicit time integration to solve the equations of motion.

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Hale [11] mentioned that one of the challenges of the explicit dynamics software is the selection of the appropriate element, in order to get an accurate solution in a short period. Most of the explicit analyses use the reduced-integration elements because they are very fast and robust. However, one of their disadvantages is the susceptibility to involve in hourglass modes. Such modes are non-physical zero-energy modes that do not generate any stress or strain. Nevertheless, they can affect solution accuracy by interfering with the real response of the structures. Hale [12] and ANSYS [13] refer that hourglass modes can be controlled by refining the mesh or by using an hourglass control algorithm. Also, it must be ensured that the hourglass energy is less than 10% of the total system of internal energy.

4 Methodology The following methodology has been proposed for the analysis of an aircraft impact on a DSC of SNF. ANSYS® code (Fig. 3) was used and the next steps were followed: (1) structural modelling with a CAD software; (2) discretization of the domain of interest; (3) the boundary conditions were introduced; (4) the solution was obtained with ANSYS® Explicit Dynamics and (5) evaluation of the results.

4.1 Structural Modelling with CATIA® (Dassault Systems) The geometric models of the DSC and the aircraft were developed with CATIA® (Dassault Systems) design software [14]. The aircraft geometry was generated with the Generative Shape Design modulus. It allows to model simple and complex shapes using wire and surface structures. On the other hand, the DSC components were designed with the Part Design modulus, in which all the components were modelled separately, and after that, the Assembly Design put them together.

Conclusions Results Solution Boundary Conditions Domain Discretization Structural Modeling CATIA® Generative Shape Design and Part Design was done

Mesh of the DSC And Aircraft Slicing by Extruding Sketches the model was made with ANSYS® Design Modeler

Fig. 3 Methodology flow chart

Applied in ANSYS® Explicit Dynamics Impact parameters: Impact at 45° at the second highest point of the container and maximum velocity of the airplane of 83.89 m/s

Explicit Dynamic High complex geometry High no linearity Explicit Solutions

Comparison of different velocities of the airplane at the same angle of impact Comparison of deformation and stresses generated by the impact Evaluation of the damage the container

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Length:

8.60 m

Wingspan:

11.20 m

Height:

2.90 m

Weight:

1158 m

Material

Density [kg/m3]

Yield Stress [Pa]

Shear Modulus [Pa]

AL5083H116

2700

1.67e08

2.692e10

Fig. 4 Geometry and characteristic of a proposed light aircraft

The models of the light aircraft and the DSC are shown in Figs. 4 and 5, respectively. In these figures, the dimensions and type of material of each component are included.

Diameter: Overall Length: Concrete Wall: Loaded Weight:

Component Bundles of waste fuel assemblies Stainless steel cylinder Rebar Concrete container

3.390 m 5.700 m 0.679 m 164631.5 kg

Materials

Density (kg/m3]

Yield Stress [Pa]

Specific Heat [J/kg °C]

Shear Modulus [Pa]

Compressive Strength fc [Pa]

Tensile Strength ft /fc [Pa]

STEEL 4340

7830

7.92e8

477

8.18

-

-

STEEL 4340

7830

7.92e8

477

8.18

-

-

STEEL 4340

7830

7.92e8

477

8.18

-

-

CONC-35MPA

2314

-

654

1.67e10

3.5e7

0.1

Fig. 5 Dry storage cask characteristics and material properties

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Fig. 6 Mesh of the dry storage cask of spent nuclear fuel and the aircraft

4.2 Domain Discretization The meshes of the aircraft and the vessel were obtained as a whole model. 13,971 hexahedral elements and 166,791 nodes were required for this purpose (Fig. 6). In accordance with ANSYS® code, the minimum mesh quality was 0.93. It was achieved through the use of global and local mesh controls, as well as the use of the “Slicing by Extruding Sketches” tool. The last one is available in the Design Modeller, in which the concrete cask was sliced in 30 bodies. The share topology mode was used.

4.3 Boundary Conditions ANSYS® Explicit Dynamics was used to simulate the impact response of an aircraft on a DSC of SNF. A short-duration problem, in which high pressures were developed, was solved. The boundary conditions for this problem were the following: the impact angle of an aircraft against the container of NSF was 45° with respect to the horizontal. The analyses were carried out considering three impact velocities, 45, 65 and 83.89 m/s. It has to keep in mind that the last one is the maximum impact velocity of the aircraft. In each simulation, all the nodes of the aircraft moved with the maximum initial impact velocity.

4.4 Results The results showed that the maximum deformation of the DSC was 3.05e−02 m, when the initial impact velocity of the aircraft was 45 m/s. On the other hand, the total deformation recorded of the DSC with an impact velocity of 65 m/s and 83.89 m/s

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was 5.09e−02 m and 7.25e−02 m, respectively. Under these conditions, a greater deformation was observed in the DSC with an impact velocity of 83.89 m/s (Fig. 7). Such deformations increase, as the impact velocity is greater. Regarding the stress field, the minimum principal stress of the DSC was − 8.13 MPa, when the initial impact velocity of the aircraft was 45 m/s. In an extreme condition (83.89 m/s), the minimum principal stress was around −12.21 MPa. The variation of such stresses during the period of analysis is shown in Fig. 8. The simulation in ANSYS® Explicit Dynamics showed the damage generated in the DSC at a maximum impact velocity of 83.89 m/s (Fig. 9). As it can be appreciated, the structural integrity of the DSC was kept after the impact and the damage zone was small. (b) (a)

Dry Storage Cask Deformation 83.89 (m/s)

65 (m/s)

45 (m/s)

Deformation [m]

8.0E-02

7.25E-02

6.0E-02

5.09E-02

4.0E-02 3.05E-02 2.0E-02

0.0E+00 1.18E-38

7.50E-03

1.50E-02

2.25E-02

3.00E-02

Time [s]

Fig. 7 a Maximum deformation of the DSC at the impact velocities, b deformation field of the DSC when the initial impact velocity was 83.89 m/s

(b) Dry Storage Cask minimum principal stress

(a) 1.18E-38

7.50E-03

1.50E-02

2.25E-02

3.00E-02

Stress [Pa]

-1.0E+06

-5.0E+06

-8.13E+06

-9.0E+06

-1.22E+07

-1.3E+07

83.89 (m/s)

65 (m/s)

45 (m/s)

Time [s]

Fig. 8 a Minimum principal stresses at the impact spot on the DSC, b minimum principal stresses field on the dry storage cask, when the initial impact velocity was 83.89 m/s

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Fig. 9 The dry storage cask damage, when it is impacted at a velocity of 83.89 m/s

Considering the extreme impact conditions (83.89 m/s) and referring to Fig. 9, it was observed in the simulation that there was no sign of cracks or damage in the rebar. The impact of a light aircraft, with a mass of 1158 kg and an impact velocity of 83.89 m/s on the DSC, does not generate permanent deformations in the concrete reinforcements. The total deformation recorded in the rebar was 1.12e−02 m (Fig. 10). On the other hand, the kinetic energy of the aircraft was converted into internal energy during impact (Fig. 11). Also, the hourglass energy was 6.85% of the internal energy (it is less than 10%). The hourglass energy is sufficiently small. This behaviour was consistent with the theoretical concepts mentioned above. Therefore, the methodology proposed in this paper is adequate to simulate the response for the light aircraft impact on a DSC of SNF.

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Fig. 10 Total deformation on the rebar, when the DSC was impacted by the light aircraft with an initial impact velocity of 83.89 m/s 5.0E+06

Internal Energy

Kinetic Energy

Hourglass Energy

Energy [J ]

4.0E+06

3.0E+06

2.45E+06

2.0E+06

1.0E+06 1.68E+05 0.0E+00 1.18E-38

7.50E-03

1.50E-02

2.25E-02

3.00E-02

Time [s]

Fig. 11 Energy interaction graph during the impact, when the DSC was impacted by the light aircraft with an initial impact velocity of 83.89 m/s

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4.5 Conclusion Based on the proposed methodology, the minimum principal stress recorded in the DSC was −12.21 MPa, when an extreme impact velocity of 83.89 m/s took place. The damage on the concrete container wall was minimum. Under these conditions, the compressive strength of reinforced concrete is 35 MPa; there is a safety factor in the DSC for SNF of approximately three. An impact simulation in explicit dynamics is an uncomplicated process if the requirements of the ANSYS® code are correctly considered and the impact geometries are generated correctly. In addition, this type of simulation is not expensive in terms of computational resources. It is important to note that the key in the development of the simulation is related to domain discretization. In particular, a mesh under conditions of global and local controls provided that the hourglass energy is below of the percentage recommended by theoretical information, generating fast and accurate results in the solution. Based on the results and the methodology used, the dry storage cask of spent nuclear fuel has a significant margin to keep the structural integrity during an aircraft impact with a maximum velocity of 83.89 m/s. Acknowledgements The authors kindly acknowledge the grant for the development of the Project 211704. It was awarded by the National Council of Science and Technology (CONACyT).

References 1. U.S.NRC.: Dry storage cask of spent nuclear fuel. Backgrounder Office of Public Affairs (2016). https://www.nrc.gov/docs/ML0622/ML062200058.pdf. Accessed 21 Mar 2018 2. EPRI 1025206.: Impacts Associated with Transfer of Spent Nuclear Fuel from Spent Fuel Storage Pools to Dry Storage After Five Years of Cooling, Revision 1, pp. 1–5. Technical Report. USA (2012) 3. U.S.NRC.: Dry Storage Cask of Nuclear Spent Fuel, Division of Spent Fuel Storage and Transportation. USA (2012). http://www.ncsl.org/documents/environ/Easton.pdf. Accessed 08 May 2018 4. Ruiz, L. M. C., Orozco, H. G.: Containers for temporary storage and transportation of spent fuel. Nuclear Safety and Radiological Protection Magazine, vol. 27, p. 9, Spain (2015). https:// www.csn.es/documents/10182/13557/Alfa+27. Accessed 020 May 2018 5. U.S.NRC.: Safety of spent fuel Transportation. NUREG/BR-0292, Rev. 2. USA (2017). https:// www.nrc.gov/docs/ML1703/ML17038A460.pdf. Accessed 28 May 2018 6. Shah, M. J.: HI-STORM dry storage cask tip-over event structural response. In: Proceedings of 18th International Conference on Structural Mechanics in Reactor Technology, ResearchGate pp. 1–11 (2005). https://www.researchgate.net/publication/242377168_HI-STORM_ DRY_STORAGE_CASK_TIP-OVER_EVENT_STRUCTURAL_RESPONSE. Accessed 2 June 2018 7. Hanifehzadeh, M., Gencturk, B., William, K.: Dynamic structural response of reinforced concrete dry storage casks subjected to impact considering material degradation. Nucl. Eng. Des. 325, 192–204 (2017). https://doi.org/10.1016/j.nucengdes.2017.10.001

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8. Siefert, A., Henkel, F., Aldoghaim, E.: Modelling of aircraft Cessna 210 for impact analysis on nuclear building. In: 23rd Conference on Structural Mechanics in Reactor Technology (2015). https://repository.lib.ncsu.edu/bitstream/handle/1840.20/33987/SMiRT23_Paper_580.pdf?sequence=1&isAllowed=y. Accessed 30 Aug 2018 9. Bangash, M.Y.H.: Structures for Nuclear Facilities: Analysis, Design, and Construction. Springer-Verlag, Berlin Heidelberg (2011) 10. ANSYS®.: Explicit Dynamics STR (2018). https://www.ansys.com/products/structures/ansysexplicit-dynamics-str. Accessed 11 Sept 2018 11. Hale, S.: Why worry about Hourglassing in Explicit Dynamics? Part 1 (2015). Available https://caeai.com/blog/why-worry-about-hourglassing-explicit-dynamics-part-i. Accessed 10 Oct 2018 12. Hale, S.: Why worry about Hourglassing in Explicit Dynamics Part 2 (2015). Available https://caeai.com/blog/why-worry-about-hourglassing-explicit-dynamics-part-ii. Accessed 10 Oct 2018 13. ANSYS®.: Workshop 01.1: Cylinder Impact, Introduction to ANSYS® Explicit Dynamics. In Customer Training Material. USA (2019) 14. Dassault Systemes.: CATIA (2019). Available https://www.3ds.com/products-services/catia/. Accessed 08 May 2018

Review of Electromagnetic Compatibility on Digital Systems of Nuclear Power Plants Israel Abraham Alarcón-Sánchez, Roberto Linares-y-Miranda, Luis Hector Hernández-Gómez, Yunuén López-Grijalba, Alejandra Armenta-Molina, Laura Guadalupe Carbajal-Figueroa and Luis Alberto Arenas-Magos Abstract The incidence of systems affected by electromagnetic interference (EMI) and radio frequency interference (RFI) in nuclear power plants has increased considerably recently, due to the replacement of analog with digital equipment. It is a consequence of the development of new technologies, which are more susceptible to interference. In addition, more attention is required to electromagnetic compatibility (EMC) in the designs, as well as the tendency to use wireless connection equipment. It has introduced more noise in the work environment. This review made an analysis of the events that have been presented in different systems related to electromagnetic interference in nuclear power plants. Such events could reduce the safety of the plant. Also, some recommendations in the electromagnetic compatibility design of instrumentation and control systems are discussed. Keywords Electromagnetic compatibility · Electromagnetic interference · Instrument and control systems · Nuclear power plant · Human–machine interface · Digital–analog systems

I. A. Alarcón-Sánchez · R. Linares-y-Miranda · L. H. Hernández-Gómez (B) · A. Armenta-Molina · L. G. Carbajal-Figueroa · L. A. Arenas-Magos Instituto Politécnico Nacional, Sección de Estudios de Posgrado e Investigación, Escuela Superior de Ingeniería Mecánica y Eléctrica, Unidad Zacatenco, “Unidad Profesional Adolfo López Mateos”, Edificio 5, Segundo Piso, Colonia Lindavista, Alcaldía Gustavo a. Madero, C.P. 07738 Ciudad de México, México A. Armenta-Molina e-mail: [email protected] Y. López-Grijalba Instituto Politécnico Nacional. Unidad Profesional Interdisciplinaria de Ingeniería, campus Hidalgo, carretera Pachuca-Actopan km 1+500, C.P. 42162 San Agustín Tlaxiaca, Hidalgo, México © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_10

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Nomenclature CRS CRT EMC EME EMI EPRI HMI IAEA IEC IEEE INPO I&C MIL-STD NPP ORNL OSS RFI SSG

Control Room System Control Room Technology Electro-Magnetic Compatibility Electro-Magnetic Environment Electro-Magnetic Interference Electric Power Research Institute Human–Machine Interface International Atomic Energy Agency International Electrotechnical Commission Institute of Electrical and Electronics Engineers Institute of Nuclear Power Operations Instrument and Control Military Standard Nuclear Power Plant Oak Ridge National Laboratory Operator Support System Radio Frequency Interference Specific Safety Guide

1 Introduction The advantages of digital technology over analog technology are evident as a result of the technological advance of the late twentieth and early twenty-first century, both in software and in hardware as they increase their performance, response speed, size, processing capacity, etc. However, these changes are still in progress at the nuclear power plants (NPPs). Some of the systems, which were designed originally, are still in use. Their designs are from the 1970s. This change has caused a discussion topic in the evaluation of the renewal of the operating license in a NPP. Despite this, the change to digital systems becomes mandatory due to the aging and obsolescence of many of the instrumentation and control (I&C) systems, which results in a nonoptimal performance of the equipment. It implies higher costs of maintenance [1]. The implementation of new technologies is given not only by digitalization but also by the implementation of optical fibers, wireless communications and artificial intelligence, which leads to the improvement of the I&C processes, as well as the

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Main control room Back-up control room

Communi -cations

Operating training programs

Emergenc y control room

CRS (Control Room System)

Operating procedures

• • • • • • • • •

Task oriented displays Intelligent alarm handling Fault detection and diagnosis Safety function monitoring Core monitoring Vibration monitoring and analysis Loose part monitoring Radiation release monitoring Condition monitoring maintenance support

New technology

(OSS, Operator Support System)

Local panels

Operating staff

Technical support centers

Fig. 1 A simplified structure of a control room system and the OSS applied in the new design of CRS in NPP considering CRT

human–machine interface (HMI). As a result, these processes are automatized; the workload decreases and human errors are minimized. However, the operator has no longer a full view of the operation of the events. He only receives results from the processed data of all the integrated systems. The term control room systems (CRS) refers to the entire human–machine interface for the nuclear stations including the main control room, back-ups control rooms and the emergency control rooms, local panels, technical support centers, operating staff, operating procedures, operating training programs, communications [2], as showed in Fig. 1. With the introduction of new technologies and the rapid evolution of digital technology, the designers were able to explore in the areas of security and operational functions. This way to the CRS was complemented with retrofittable operator support system (OSS) (see Fig. 1). With the addition of new technology applied to control room CRT (control room technology), a new trend has begun, including topics related to artificial intelligence, neural networks, fuzzy logics, cognitive user models, etc. [2]. The EMC can provide the necessary security, regarding the analog–digital technological evolution, since it refers to the coexistence of everything that works with electrical energy, such as sources, transmission lines, radiofrequency, power equipment, motors, etc. In an aggressive electromagnetic environment of nonionizing energy, whose effects are manifested in EMI, and malfunction of the equipment can be generated. At the same time, the safety of the plant can be compromised. Part of the evolution of the current control systems in nuclear devices is presented in [3], where it also provides an explanation of how cybersecurity emerged, with a concern in its reliability and predictability. The performance and normativity of the I&C systems of the NPP, with respect to the EMI as individual elements, are evaluated in accordance with the EMC. The related methods, according to the regulatory guide 1.180 Revision 1 of the Nuclear Regulatory Commission of the USA, are established according to the procedures

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specified by military standard MIL-STD 461. They are suitable to all plant equipment. Both documents are directly and indirectly related to safety and specify the full frequency coverage of the EMI sources, RFI from 30 Hz to 10 GHz. However, these guides do not consider the systems in operation as a whole. In addition, there is no information on the electromagnetic environment in the control room or in any other place of the plant in general, which would be a topic to be carried out with an analysis of the instrumentation and the sensors as modules. The revolution of wireless systems in the nuclear sector has the ability to provide operating solutions in places of difficult access or in the reduction of the cables by their wireless transmission. In this way, the aging of materials in the wiring systems is avoided, as well as the costs that are related to this matter [4]. Despite this, the nuclear sector is cautious in using them, due to the rigorous regulations related to security and reliability. The issues of cybersecurity or problems related to EMI put into discussion its implementation, as well as one of the main concerns and limitations is the useful life of the batteries that these systems could have. Nonetheless, the tendency and interest to use them are increasing in the nuclear industry [5]. So, researchers are conducting a series of investigations, such as the analysis of the systems that transmit at intervals to preserve as much as possible a charge of approximately two years, as well as systems that collect their own energy (battery-free) [6]. The EMI/RFI as well as the EMC in NPP has been treated in several papers. Oranchak and coworkers [7] carried out studies and discussed recommendations of the effects of RFI in nuclear plants in 1980. In accordance with the standards from that time (IEEE-472, Standard PMC 33.1). Cirillo [8] refers to the lack of attention in the EMC in the design of the I&C systems. Despite this, many of these systems continue their operation and the suggested recommendations indicate exclusion zones. A guide for EMC in the NPPs is given in [9], according to the research carried out by the Oak Ridge National Laboratory (ORNL). The concept of the EMC has not changed since its original definition. However, the evaluation procedures can be modified, since one of the essential problems in the digital part is directly associated with the processes of guided and radiated signals. A second point is related to the integrity of the signal and a third one is to the electrical systems. This paper discusses the basic problems to be considered in EMC from a global point associated with digital technologies that are given by the integrity of the signal and the trends of wireless sensors.

2 Human–Machine Interface in Nuclear Power Plant The term HMI does not only involve the specific interaction between the user and the computer interface but also refer to the design of any system or tool, for example, the design of an application of a mobile device, the design of robust environments such as the control room of a NPP or the design of a complex virtual system. Thus, it must be understood by “machine” everything that can be handled by humans and its relationship being known as “interface” [10]. For the nuclear sector, the operating

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personnel must interact with the I&C systems, which include the screens, controls and interface with the support system of the operator [11]. The HMI increases the efficiency of NPPs, when the I&C systems are renewed. However, techniques and methods must be established to support the design of the interface with the operating personnel of the plant. Similar considerations have to be taken with the optimization and specific designs due to the environment in which the digital systems of the plant will operate. All these aspects make easier the design of an interface of I&C systems for the operative personnel, especially with those systems that are extremely complex. In the last situation, the interface with the operative personnel could be compromised, since it is not an adequate design for it. The design must take into account also the coexistence of high and low energy on systems in a NPP. In this context, it is essential that in the nuclear sector, the I&C systems have to be rigorously designed and validated by the operating personnel, because some of these systems are committed to the plant safety. Under this scheme, some systems or devices controlled by complex interfaces, which have been applied in sensitive systems in the industry, have the possibility (and in some cases has occurred) of presenting serious consequences in the interface [12]. Thus, the designers of the new I&C systems have to consider that the HMI must be designed together with the operators of the plant because the conception of the system can be different from the original idea. Therefore, operator–designer communication provides levels of knowledge necessary for the designs of the HMI. This is how rigorous methods and regulations have been established to guide the interface design process in which error is minimized. The productivity is increased by facilitating the use and management of systems, programs, devices and applications in all areas of the industrial activities [12]. With the development of the HMI in the establishment of standards and in the design of the interfaces, a need has arisen to involve different concepts and techniques based on multiple disciplines, since the interaction of the machine and the man consists of understanding the operation of one as the other. In the operation of NPPs, the positive characteristics found in the reference designs should be retained, as well as problems arising from a poor operational experience have to be avoided. Therefore, the design of the I&C systems in the HMI must apply the principles of defense in depth [11]. Thus, the transition of existing I&C systems in NPPs to the new generation requires deep analysis from the perspective of engineering human factors, so that they can adapt to the operation of the equipment. Therefore, an increase in errors due to poor handling by the operator and maintenance personnel has to be avoided with an adequate training of personnel. In this way, errors due to poor handling by the operator and maintenance personnel are reduced, once that the changes have been made. In the design of modern control rooms in the nuclear plants, the visualizations of the safety parameters and the functions of the accident monitoring system in the HMI are already integrated. However, training can sometimes be limited to a specific situation, so it is essential that all scenarios have to be considered, including the starting, the shutting down and accident event of the reactor.

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Based on the above concepts, the International Atomic Energy Agency (IAEA) has published the specific safety guide for the design of instrumentation and control systems for nuclear plants No. SSG-39. It establishes the following series of requirements in the HMI, which have been included in its 2016 edition [11]: (a) Must, as far as possible, accommodate the different roles and responsibilities of the many types of operating personnel that are expected to interact with the systems. (b) Must be designed with primary attention to the role of the operator responsible for the safe operation of the equipment. (c) It should support the development of knowledge of the common situation by the crew of the control room, for example. Through large plant status screens mounted on the wall. (d) Must provide an effective overview of the state of the plant. (e) To the extent possible, it should apply the simplest design that is compatible with the requirements of functions and tasks. (f) Must be designed to minimize dependence on operator training. (g) They must present information in a way that operators can quickly recognize and understand it. (h) It should be adapted to the failure of the analog and video screens without significant interruption of the control actions. (i) It must reflect the consideration of human physiological characteristics, the characteristics of human motor control and anthropometry. Regarding the regulations, the safety specific guide No. SSG-39 [11] indicates that for HMI, the following considerations, shown in Table 1, must be taken into account. So the upgrading of I&C systems is carried out, taking into account the recommendations of the specific Safety Guide No. SSG-39 [11] in the design of the HMI and the Regulatory Guide 1.180 Revision 1 [13] for EMC. It has to be a priority in the design due to the environment to which the equipment is subjected, coupled with the innovations of wireless systems in the nuclear industry. Table 1 Relationship between HMI and international standards [11] Considerations

Internationally used instrumentation and control standards

Control rooms

IEC 60964, IEC 61772, IEC 62241, IEEE 576

Additional control rooms

IEC 60965

Accident monitoring

IEEE 497

General principles related to the engineering of human factors for I&C systems

IEC 61839, IEC 61772, IEEE 1023, IEEE 1082

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3 Upgrade of the I&C Systems in Nuclear Power Plants The accelerated development of digital technology makes systems obsolete quickly. Therefore, the NPPs have to upgrade their equipment to digital systems or, where appropriate, to advanced analog systems. However, this new technological age makes use of a combination of different technologies and the use of wireless systems is increasing. These technologies have the possibility of being more susceptible to EMI/RFI environments than the existing systems in those plants because the manufacturers of digital systems incorporate increasingly, higher clock frequencies and lower logical level voltages in their designs. It means that there is a greater probability that the EMI causes a misinterpretation of the signal by passing them through legitimate logical signals. However, the benefits that can be obtained with the digital I&C systems are the following: 1. The number of the analog circuits necessary for measuring the I&C decreases. As a result, there are fewer circuits that are committed to a process and they could be reduced victims of EMI. The data obtained from the measurements is more precise or accurate. Besides, it can be refined through digital data processing programs. 2. It is possible to modify the measurement parameters more easily in digital systems than in analog systems. It makes them more versatile systems and adaptable to changes in the plant. In general, making changes in hardware to software functionality implies faster component installation and more economical maintenance. 3. The I&C systems are miniaturized and reduced and the process is enhanced through their integrated circuits. The overall size of the devices decreases; thus, the space required for their installation is smaller. As a result, the wiring for the transmission is reduced, the costs are lower and there is available higher transmission speed. Besides, the tendency to use wireless for the transmission of data has increased. 4. The digital technology increases the processing capacity. So, more operations can be done simultaneously. This is a great advantage over the analog systems. 5. The possibility to automate processes is increased, which leads to decrease in human error. 6. The I&C systems can perform diagnostics of their own systems. It reduces maintenance costs and improves the monitoring of the functionality and reliability of the equipment. The I&C systems, regardless of whether they are analog or digital system, can fail in the operation. Such failures are classified into two modes: (1) The I&C systems do not fulfill their functions in which the plant safety is compromised. (2) Spurious signals that can cause false readings are generated. As a result, the protection systems can be activated without any reason. The first case is considered more critical because the integrity and safety of the plant are compromised. The second can not be so critical, since a false reading may or not trigger a potentially dangerous situation of

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the plant. These false readings may be caused by the EMI. This is the importance of the EMC in the design of the I&C systems. A nuclear plant requires more than 10,000 I&C systems. They vary in their function, hardware, software and HMI. A generalization or prediction of their failure can not be established. At the same time, it is not feasible to evaluate each one of the I&C systems. Therefore, the evaluations have been focused on the systems that are committed to plant safety [14]. Despite the function of the digital I&C systems and according to the study carried out by Electric Power Research Institute (EPRI) in 2011, the reported events, in which the EMIs have been involved, have increased since 2000. From one to five events have been reported each year by the system of the Institute of Nuclear Power Operations (INPO). 32 events related to EMI were reported. The I&C systems were involved in 16 of those events. In addition to this, 45 events were related to electrostatic discharges [15]. Some of the events have been related to the plant safety. However, rapid reactions of the reactor, due to the false readings, were taken. It is also reported the increase of cases related to RFI. They have been attributed to digital devices. The new simulation technologies allow the study in places or situations of difficult access, such as the electromagnetic environment of a nuclear power plant. These technologies are increasingly relevant in all aspects of engineering. In case of NPP environment can be simulate the electromagnetic wave propagation. These technologies have the possibility of solving or explaining the EMI that takes place in a NPP. This event can take place in the operation of the plant or it has to be considered in the design of the I&C systems.

4 Electromagnetic Compatibility The I&C systems, in which the NPP safety is compromised, must comply with the EMC standards. At least, the following situations have to be considered. 1. Radiated: Immunity and emissions. Equipment must comply with a standard of immunity related to radiated emissions. An electromagnetic field, capable of causing an EMI, must not be generated. 2. Reliable information. The information that is transmitted, received and interpreted by the I&C systems must not be changed due to uncoupled cables or altered by EMIs. 3. Damage caused by fast transients. They must be capable of supporting electrostatic charges that can modify the sensors information and I&C equipment. 4. Harmonic distortion. Harmonic voltage variations that affect signal quality should be avoided. These considerations have been considered in the MIL-STD 461 [16] of the military standards and the IEC-61000 series [17] of the international standards. They establish the immunity requirements, which are required for a proper functioning

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of the equipment. They consider the cases of radiated and conducted signals and issues such as: fast transients, voltage drops, electromagnetic field by radiofrequency, damped oscillatory disturbances, harmonics distortion, among others. A classification of the electromagnetic environment (EME) and the level of immunity of the I&C systems in NPP has been discussed in [3]. It is in accordance with the area in which the equipment is installed and the EME. Each element is evaluated according to the applicable EMC regulations. Functional quality criteria of the I&C systems of the NPP are divided into three categories, depending on the operational quality during the tests of immunity to disturbances [3]: A. Systems must continue to operate as planned. Both the performance and the loss of the function of the equipment must not exceed the limit established by the manufacturer when used as intended. Therefore, a permissible loss of performance is allowed while it does not exceed the limits established by the manufacturer. If the manufacturer does not specify a minimum level of performance or allowable loss of this, can be obtained according to the description and documentation of the product and what the user expects from it to be used reasonably as intended. B. The systems will continue to function as planned after the tests. Both the performance and the loss of the function of the equipment must not exceed the limit established by the manufacturer when used as intended. Therefore, a permissible loss of performance is allowed while it does not exceed the limits established by the manufacturer. However, performance degradation is allowed during the test. If the manufacturer does not specify a minimum level of performance or allowable loss of this, can be obtained according to the description and documentation of the product and what the user expects from it to be used reasonably as intended. C. Temporary loss of the system functions is allowed, provided that the function is self-recoverable or can be restored through the operation of the controls.

5 Conclusion All the electromagnetic radiation phenomena that cause EMI levels of immunity and the transmission of reliable information without problems of disturbances are considered by the standards related to EMC in NPPs. The manipulation of all the equipment in its operational conditions, in the installation processes and in electrostatic discharges, is important and this is a point that should be given attention in all the I&C systems. In this scenario, the voltage variation, the generation of harmonics, the susceptibility, protection, among others must be avoided. Regarding the pulses of high energy, more research is required. Some transients can take place in which safety can be compromised in industrial process, including the NPPs. Actually, new technologies are available. The introduction of digital systems in NPPs is not completely mature, because this technology is in constant evolution.

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This is a dilemma because updating is required, since the obsolescence of analog and some digital systems is a fact. Diverse procedures are required to predict or generate certainty of the levels of nuclear safety required. In this respect, simulation is a tool which is required for this purpose. The number of devices, equipment and electronic systems in a NPP control room is sufficient. Each of them is operating individually. They have been evaluated to meet the compliance of the electromagnetic compatibility in accordance with the rules and procedures available. However, despite complying with the regulations, each of the devices emits a level of electromagnetic radiation to the environment. It is increased by the superposition of electric field. An electromagnetic environment more complex is generated. It is not usually evaluated and could exceed the limits toward exposure, not only for the equipment but also for the personal of operation. Acknowledgements The authors kindly acknowledge the grant for the development of Project 211704. It was awarded by the National Council of Science and Technology (CONACYT). Statement The conclusions and opinions stated in this paper do not represent the position of the National Commission on Nuclear Safety and Safeguards, where the commission is active part of the project mentioned in the acknowledgment. Although special care has been taken to maintain the accuracy of the information and results, all the authors do not assume any responsibility for the consequences of its use. The use of particular mentions of countries, territories, companies, associations, products or methodologies does not imply any judgment or promotion by all the authors.

References 1. Yoon, M.-H., et al.: Adopting modern computer system technology to nuclear power plant operations. In: IFAC Distributed Computer Control Systems, Seoul, Korea (1997). https:// www.sciencedirect.com/science/article/pii/S1474667017426802 2. Manninen, T.: Process monitoring systems of Loviisa Nuclear Power Station, Control Room Systems Design for Nuclear Power Plants. In: IAEA-TECDOC-812, IAEA, Vienna (1995). https://inis.iaea.org/search/search.aspx?orig_q=RN:27002052 3. Fu, M., et al.: Research on electromagnetic compatibility and electronic compatibility standard of instrument control equipment in nuclear power plant. In: International Forum on Energy, Environment and Sustainable Development (IFEESD). Shenzen, China (2016). https://dx.doi. org/10.2991/ifeesd-16.2016.199 4. Hashemian, H.M.: Nuclear power plant instrumentation and control. In: Tsvetkov, P. (ed.) Nuclear Power—Control, Reliability and Human Factors. Texas A&M University, United States of America (September 2011) 5. Laikari, A., et al.: Wireless in nuclear. In: Nuclear, Energiforsk Nuclear Safety Related I&CENSRIC. Report 2018:513. Switzerland (July 2018) 6. Nekoogar, F., Dowla, F.: A robust wireless communication system for electromagnetically harsh environments of nuclear facilities. In: NPIC&HMIT 2017, San Francisco, California, USA (June 2017). https://pdfs.semanticscholar.org/b054/ 7a5aa4304c4f401fd74ef1ab70b289008319.pdf?_ga=2.102429255.799180829.15716861531885068037.1571686153 7. Oranchak, J.R., et al.: RFI effects on nuclear power plant I&C equipment. In: IEEE Transactions on Nuclear Since, San Francisco, California, USA (1979). https://doi.org/10.1109/TNS.1980. 4330941

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8. Cirillo, J., et al.: Electromagnetic compatibility in nuclear power plants. In: IEEE Transactions on Nuclear Science, Oak Ridge, TN, USA (February 1986). https://doi.org/10.1109/TNS.1986. 4337268 9. Wood, R.T., et al.: Electromagnetic compatibility in nuclear power plants. In: Proceedings of the International Conference on Future Nuclear Systems: GLOBAL. USA (1999). https://inis. iaea.org/search/search.aspx?orig_q=RN:30048037 10. Asociación Española de Normalización y Certificación (AENOR): UNE 139802:2003. Aplicaciones informáticas para personas con discapacidad. Requisitos de accesibilidad al ordenador. AENOR, Madrid, Spain (2013) 11. IAEA: Design of Instrumentation and Control Systems for Nuclear Power Plants. IAEA Safety Standards Series No. SSG-39. Vienna, Austria (2016) 12. Ezquerra, N., et al.: Interacción hombre-máquina y usabilidad: Diseño centrado en el usuario. ResearchGate. Spain, (November 2015). Available from: https://www.researchgate.net/ publication/266358347_Interaccion_hombre-maquina_y_usabilidad_Diseno_centrado_en_ el_usuario 13. Guide R 1.180 Revision 1: Guidelines for Evaluating Electromagnetic and Radio-Frequency Interference in Safety-Related Instrumentation and Control Systems. USA (2003) 14. Gehl, A.C., et al.: Aging Assessment of Reactor Instrumentation and Protection System Components NUREG/CR-5700. Oak Ridge National Laboratory, USA (1992) 15. EPRI: Assessment of Electromagnetic Interference Events in Nuclear Power Plants. 2011 Technical Report, USA (December 2011) 16. Department of Defense of USA: MIL-STD-461G. Requirements for the Control of Electromagnetic Interference Characteristics of Subsystems and Equipment, USA (March 2015) 17. International Electrotechnical Commission: Structure of IEC 61000. Geneva, Switzerland (November 2018). Available from: https://www.iec.ch/emc/basic_emc/basic_61000.htm

Effect of Beam Rigidity on the Lateral Stiffness of a One-Storey Frame Meziane Chalah, Farid Chalah, Salah Eddine Djellab and Djillali Benouar

Abstract The present work deals with the effect of beam to column rigidity ratio on the lateral stiffness of a frame. This investigation is relative to the case of a one-storey one-bay frame. The displacement method is used to formulate the global stiffness matrix of the frame. The treated problem consists of applying a lateral force at the storey to obtain its displacement and deduce the lateral stiffness. It is found that the lateral stiffness of the frame is greatly influenced by the beam to column rigidity ratio. A high value of the latter makes the overall stiffness corresponding to the sum of each column’s stiffness. For intermediate values, the influence of the beam becomes important and must be considered in the structural analysis. For very lower values, the columns behave as hinged at their top. The provided plotted curves allow finding the lateral stiffness for different base boundary conditions of the columns. Keywords Lateral stiffness · Column stiffness · Beam rigidity · Single-span · Lateral force

1 Introduction The analysis of the lateral stiffness of a single-storey single-bay portal frame constituted an evident subject in the analysis of structures using either laboratory experimentation [1] or structural analysis [2]. In this context, a research work presented approximate methods to evaluate storey stiffness and interstorey drift of RC buildings in seismic area [3]. Another research development proposed a method [4] for estimating the lateral stiffness of building systems where it considers an equivalent single-bay single-storey frame module for every storey of the real multi-bay multistorey frame. Also, the lateral stiffness estimation in frames with its implementation to continuum models for linear and nonlinear static analysis was investigated in [5]. The adopted approach in this study consists of considering the resisting elements constituted by the two columns and the supported beam in the global stiffness. In M. Chalah (B) · F. Chalah · S. E. Djellab · D. Benouar Faculty of Civil Engineering, Usthb, Algiers, Algeria © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_11

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general, this problem is treated to evaluate the bending moment, shear force and normal force diagrams in the different parts of the structures. But, in the context of the present study, the problem is tackled to find the lateral displacement under a lateral force. This latter is applied as two forces F/2 at the two linking joints between the beam and the two columns. After constructing the stiffness matrix, the problem is solved for the lateral displacement. This latter allowed expressing the global lateral stiffness as a function of the beam to column rigidities ratio.

2 Analysis of a One-Storey One-Bay Frame with an Infinitely Rigid Beam Three situations appear in the case where the beam is considered infinitely rigid, as follows: 1. The bases of the columns are fixed: the lateral displacement and the rotation at the base of each column are restrained (fixed) 2. The first column (or the second one) base is free to rotate (hinged or pinned) and that of the second base is fixed (clamped) 3. The two bases rotations are released (hinged) while the lateral displacements are restrained (locked). Figures 1, 2 and 3 given in Sects. 2.1–2.3 show these different possible classical situations. Fig. 1 Representation of one-storey one-bay frame with an infinitely rigid beam. Case where the lateral stiffness is ensured by two fixed–fixed columns

Fig. 2 Representation of one-storey one-bay frame with an infinitely rigid beam. The rotation at the base of the first column is released

Infinitely rigid beam

K2= H

K1=12E.I1/H3

K1=12E.I1/H3

Infinitely rigid beam K2=

K1,PF=3E.I1/H3

K1,FF=12E.I1/H3

H

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Infinitely rigid beam

Fig. 3 Representation of one-storey one-bay frame with an infinitely rigid beam. The ends restraint conditions of the two columns are hinged-fixed (pinned-fixed)

K= K1,PF=3E.I1/H3 K1,PF =3E.I1/H

H 3

2.1 Two Fixed–Fixed Columns with an Infinitely Rigid Beam The generally considered system in structural dynamic analysis is constituted by a one-storey one-bay frame with an infinitely rigid beam as shown in Fig. 1. The stiffness of a fixed–fixed column is: K FF = 12

E.I1 H3

(1)

where: I 1 Inertia of a column. H Total column height. E Young modulus of the used material. The contribution of the two columns stiffness is:   E.I1 K T = 2 · 12 3 H

(2)

or: K T = 24

E.I1 H3

(3)

This stiffness value corresponds to the case of two fixed–fixed columns.

2.2 A Hinged-Fixed Column with a Fixed–Fixed Column with an Infinitely Rigid Beam In this case, the rotation of the first column base is released, as given in Fig. 2. Thus, the column stiffness in this situation where the ends’ conditions are hingedfixed (pinned-fixed) will be:

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E.I1 H3

(4)

E.I1 E.I1 +3 3 3 H H

(5)

K PF = 3 and then, the total stiffness becomes: K T = 12

that gives for the case where one column is hinged-fixed and the second fixed–fixed, a stiffness value of: K T = 15

E.I1 H3

(6)

2.3 Two Hinged-Fixed Columns with an Infinitely Rigid Beam In this subsection, the two bases of the columns in Fig. 3 are considered hinged. As K PF = 3E.I 1 /H 3 , the total stiffness in the case where the ends’ conditions of the columns are hinged-fixed or pinned-fixed is: KT = 6

E.I1 H3

(7)

3 Analysis of a One-Storey One-Bay Frame with a Flexible Beam The present development concerns the analysis of a one-storey one-bay frame. The two columns are considered rigidly linked to the beam they support. The characteristics of the beam are I 2 and L corresponding to the inertia and the length, respectively.

3.1 Analysis of a Frame in the Case Where the Columns Are Fixed at Their Bases Figure 4 shows the situation where the two bases are fixed.

Effect of Beam Rigidity on the Lateral Stiffness … Fig. 4 Representation of a one-storey one-bay frame with a beam rigidly linked to the columns. Case 1: the two columns bases are fixed

119

F/2

F/2

beam 3

4

E.I2

E.I1

E.I1

H

Column2

Column1

2

1 L

Applying the concept of structural analysis allows formulating the problem to be solved to find the lateral displacement of the beam and the two rotations at its two ends. The equations of equilibrium are: M31 + M34 = 0

: Node 3

(8)

M43 + M42 = 0

: Node 4

(9)

t31 + t42 = 0

(10)

where: M Bending moment. t Shear force. The bending moments and shear forces at the ends of the column element 1–3 are expressed as follows: ⎧ ⎫ ⎡ 2E.I1 M13 ⎪ ⎪ H ⎪ ⎪ ⎨ ⎬ ⎢ 4E.I 1 M31 H =⎢ 6E.I1 ⎣ ⎪ t ⎪ ⎪ H2 ⎩ 13 ⎪ ⎭ −6E.I1 t31 H2

−6E.I1 H2 −6E.I1 H2 −12E.I1 H3 12E.I1 H3

⎤

  ⎥ θ3 ⎥ ⎦ Z3 = Z

where: θ Rotation of column end section. Z 3 Lateral displacement of the left node of the beam. Z Lateral translation of the beam.

(11)

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Beam 3–4: ⎧ ⎫ ⎡ 4E.I2 M34 ⎪ ⎪ L ⎪ ⎪ ⎨ ⎬ ⎢ 2E.I 2 M43 L ⎢ = ⎣ 6E.I 2 ⎪ t ⎪ ⎪ L2 ⎩ 34 ⎪ ⎭ −6E.I2 t43 L2

2E.I2 L 4E.I2 L 6E.I2 L2 −6E.I2 L2

⎤

  ⎥ θ3 ⎥ ⎦ θ4

(12)

  ⎥ θ3 ⎥ ⎦ Z4 = Z

(13)

Column element 2–4: ⎫ ⎡ 2E.I1 ⎧ M24 ⎪ ⎪ H ⎪ ⎪ ⎬ ⎢ 4E.I ⎨ 1 M42 H ⎢ = ⎣ 6E.I 1 ⎪ t ⎪ ⎪ H2 ⎭ ⎩ 24 ⎪ −6E.I1 t42 H2

−6E.I1 H2 −6E.I1 H2 −12E.I1 H3 12E.I1 H3

⎤

where: Z 4 Lateral displacement of the right node of the beam. The problem to be solved to find θ 3 , θ 4 and Z is expressed in Eq. 14, under the form of stiffness matrix, displacements and forces vectors. ⎡ 4E.I1 ⎣

H

+

4E.I2 L

2E.I2 L 1 −6 E.I H2

2E.I2 L 4E.I1 4E.I2 + L H E.I1 −6 H 2

⎤⎧ ⎫ ⎧ ⎫ 1 −6 E.I ⎨ θ3 ⎬ ⎨ 0 ⎬ H2 E.I1 ⎦ −6 H 2 θ4 = 0 ⎭ ⎩ ⎭ E.I1 ⎩ 1 12 E.I + 12 Z F 3 3 H H

(14)

The determination of the lateral displacement Z is as follows: ⎛⎡ 4E.I1 Det⎝⎣ Z=

⎛⎡ Det⎝⎣

H

+

⎤⎞ 2E.I2 0 L 4E.I2 1 + 4E.I 0 ⎦⎠ L H E.I1 −6 H 2 F ⎤⎞ 2E.I2 1 −6 E.I L H2 4E.I2 1 1 ⎦⎠ + 4E.I −6 E.I L H H2 E.I1 E.I1 −6 H 2 24 H 3

4E.I2 L

2E.I2 L 1 −6 E.I H2 4E.I1 2 + 4E.I H L 2E.I2 L 1 −6 E.I H2

(15)

Replacing E.I 1 /H by α 1 and E.I 2 /L by α 2 gives:    4α1 + 4α2 2α2 0    2α2 4α1 + 4α2 0    −6 α1 −6 αH1 F H

Z=  4α1 + 4α2 2α2 −6 Hα12   2α2 4α1 + 4α2 −6 Hα12   −6 α1 −6 Hα12 24 Hα13 H2

     

(16)

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The development of the numerator is as follows:   F (4α1 + 4α2 )2 − (2α2 )2

(17)

  F 16α12 + 32α1 α2 − 16α22

(18)

That gives:

The development of the denominator is summarized in Eqs. 19 and 20:   4α + 4α −6 α12 H (4α1 + 4α2 ) 1 α1 2 −6 H 2 24 Hα13   α1  2α2 −6 Hα12  − 6 2  H 4α1 + 4α2 −6 α12 

  α1    − 2α2  2α2 −6 H 2 α α   −6 12 24 13 H H

    (19)

H

96

α13 α 2 α2 α1 α 2 + 624 1 2 + 288 21 2 H H H

(20)

Z takes the form of Eq. 21: Z=

  F 16α12 + 32α1 α2 − 16α22 α3

96 H12 + 624

α12 α2 H2

+ 288

α1 α22 H2

(21)

Considering that stiffness is the force to apply in order to obtain a unitary displacement, the global stiffness of the studied frame is: KT =

α1 96α12 + 624α1 α2 + 288α22 H 2 16α12 + 32α1 α2 + 12α22

(22)

By dividing both the numerator and the denominator by α 21 and introducing the beam to column rigidity ratio r = α 2 /α 1 = (I 2 /L)/(I 1 /H), it is rewritten after simplifications under the form of Eq. (23). K T = 12

α1 2 + 13r + 6r 2 H 2 4 + 8r + 3r 2

(23)

After finding the roots of the two expressions in the numerator and the denominator, a contracted relationship is formulated: K T = 12

E.I1 6r + 1 H 3 3r + 2

(24)

As the reference is the total columns’ stiffness, the relationship giving the global stiffness in as a function of the beam to column rigidity ratio becomes:

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K T = 24

E.I1 (6r + 1) . H 3 (6r + 4)

(25)

which defines a new expression, as follows: K T = 24

E.I1 .F(r ) H3

(26)

F(r ) =

(6r + 1) (6r + 4)

(27)

with:

where: F(r) is the dimensionless factor that constitutes the correction to give to the total stiffness in the case of a flexible beam in function of the rigidities ratio in the situation where the columns’ bases are fixed. Figure 5 shows the variation of the dimensionless correcting factor F(r) as a function of the beam to column rigidities ratio in the case the two columns bases are fixed. The frame lateral stiffness in the case of a very rigid beam (r ≈ 100) is essentially governed by the two columns. For a beam to column rigidities ratio, r greater than 10 gives at least a lateral stiffness of about 0.95. An r value comprised between 10 and 0.1 induces more variation of the lateral stiffness. These states correspond to various frames configurations. For the particular situation of equal rigidities

Fig. 5 Variation of the dimensionless correcting factor in function of the rigidities ratio in the case of columns with fixed bases

Effect of Beam Rigidity on the Lateral Stiffness …

123

(r = 1), the frame lateral stiffness is of about 0.7 of the same frame having an infinitely rigid beam. This indicates the importance of the beam rigidity effect on the overall behaviour of frames of one-storey one-bay. As expected, for very lower values of the ratio r characterizing a relatively weak beam, the dimensionless correcting factor tends to a limit value of 1/4 = 0.25. Such a beam linking the two columns plays the role of a hinging device that makes each column behaviour becoming fixedhinged. Thus, each column stiffness takes a value of 3E.I 1 /H 3 and then the total frame lateral stiffness will be 6E.I 1 /H 3 .

3.2 Analysis of a Frame in the Case of One Column Base Is Hinged and the Second Is Fixed Figure 6 shows the system with a column having a hinged base in association with another fixed at its base. The analysis of such a statically indeterminate structure allowed to express the problem to be solved under matrix form as given in Eq. 28: ⎡ 3E.I1 ⎣

H

+

4E.I2 L

2E.I2 L 1 −3 E.I H2

2E.I2 L 4E.I2 1 + 4E.I L H E.I1 −6 H 2

⎤⎧ ⎫ ⎧ ⎫ 1 −3 E.I ⎨ θ3 ⎬ ⎨ 0 ⎬ H2 1 ⎦ = 0 −6 E.I θ 2 H ⎩ 4⎭ ⎩ ⎭ 1 15 E.I Z F 3 H

(28)

Similar developments to the previous section are conducted to obtain Eq. 29: KT = 3

E.I1 15r 2 + 26r + 3 H 2 3r 2 + 7r + 3

(29)

As the total stiffness in the case of a high ratio is equal to 15E.I/H 3 , it becomes more practical to rewrite the previous relationship as follows: K T = 15

Fig. 6 Representation of a one-storey one-bay frame. Case 2 where the first column base is hinged and that of the second is fixed

E.I1 15r 2 + 26r + 3 H 2 15r 2 + 35r + 15 F/2

(30)

beam

F/2

E.I2 E.I1

E.I1

Column 1

Column 2 L

H

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Fig. 7 Variation of the rigidities ratio of frame in the case of a first column base hinged and that of the second fixed

Figure 7 gives the dimensionless correcting factor F(r) as a function of r in the case where the first column base is hinged while that of the second is fixed. In the case of very low values of r, the lateral resistance of the frame is only ensured by the presence of the right column. The stiffness of the latter takes a value of 3E.I 1 /H 3 that characterizes a fixed-hinged column.

3.3 Analysis of a Frame in the Case Where the Columns Are Hinged at Their Bases Figure 8 represents a frame in the particular case where the bases of the two columns are hinged. Fig. 8 Representation of a one-storey one-bay frame with a beam rigidly linked to the columns. Case 3: two hinged bases

beam

F/2

F/2

E.I2 H E.I1

E.I1 Column 1

Column 2 L

Effect of Beam Rigidity on the Lateral Stiffness …

125

Adopting the same procedure as carried out before allows expressing a relationship as written in Eq. 31 giving the corresponding lateral stiffness K: KT = 6

E.I1 2r . H 3 (2r + 1)

(31)

The dimensionless correction factor F(r) value tends to the unity (1) for high values of the ratio and then, the global lateral stiffness as waited takes the value of: KT = 6

E.I1 H3

(32)

It corresponds to the situation of a frame with an infinitely rigid beam. Figure 9 represents a plot of the dimensionless correcting factor F(r) for the case where of two hinged columns bases. The main remark is relative to very low values of the rigidities ratio where the two columns behave as hinged at their top. The correction factor tends to a nil value (zero). The resulting effect is a very weak global stiffness since the system is nearly a mechanism. Comment For relatively high beam rigidity, the frame behaviour will be as that previously introduced in the first section (presence of an infinitely rigid beam). But for intermediate values of the ratio, a correcting function is derived for each situation of the columns base condition to evaluate the global lateral stiffness without a need of structural problem solving.

Fig. 9 Variation of the rigidities ratio of the frame in the case of two hinged columns bases

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4 Conclusion The analysis of a one-storey one-bay frame may be treated easily in the case of an infinitely rigid beam. However, a particular attention must be accorded in the case where the beam is not infinitely rigid. For an extreme situation of a very weak beam, two main situations are to be distinguished, namely at least one column base is fixed and two columns having their bases hinged. In the first case, the dimensionless correcting factor value is: 1/5 corresponding to the case where the lateral stiffness is only ensured by the right column having its base initially fixed that becomes as a column fixedhinged for a stiffness of 3E.I 1 /H 3 while the left column stiffness becomes nil (hinged–hinged column). 1/4 that shows a total stiffness dropping from 24E.I 1 /H 3 to 6E.I 1 /H 3 . The two columns having their bases initially fixed behave as hinged at their top and then each column stiffness becomes 3E.I 1 /H 3 . For the second case of hinged bases, the beam weakness makes the global behaviour changing from a frame to a no braced truss (without a diagonal) becoming as a mechanism.

References 1. Opinion, J., Rico, V., Guardado, M., Boyajian, D., Zirakian, T.: SEE laboratory: single-story, single-bay portal frame. J. Civ. Eng. Arch. 11 (2017). https://doi.org/10.17265/1934-7359/2017. 05.004 2. Megson, T.H.G.: Structural and Stress Analysis. Arnold [u.a.], London (1996) 3. Caterino, N., Cosenza, E., Azmoodeh, B.M.: Approximate methods to evaluate storey stiffness and interstory drift of RC buildings in seismic area. Struct. Eng. Mech. 46, 245–267 (2013). https://doi.org/10.12989/sem.2013.46.2.245 4. Hosseini, M., Imagh-e-Naiini, M.R.: A quick method for estimating the lateral stiffness of building systems. Struct. Des. Tall Build. 8, 247–260 (1999). https://doi.org/10.1002/(SICI)10991794(199909)8:3%3c247:AID-TAL126%3e3.0.CO;2-K 5. Ero˘glu, T., Akkar, S.: Lateral stiffness estimation in frames and its implementation to continuum models for linear and nonlinear static analysis. Bull. Earthq. Eng. 9, 1097–1114 (2011). https:// doi.org/10.1007/s10518-010-9229-z

Fundamental Transverse Vibration Circular Frequency of a Cantilever Beam with an Intermediate Elastic Support Lila Chalah-Rezgui, Farid Chalah, Salah Eddine Djellab, Ammar Nechnech and Abderrahim Bali

Abstract The problem of analyzing the dynamic vibration of a uniform beam was widely investigated in the scientific literature where various numerical methods were considered. The objective of the present study is to investigate the fundamental transverse vibration circular frequency ω1 of a cantilever beam with an intermediate elastic support of variable abscissa a. The analysis is based on the Euler–Bernoulli assumptions and carried out by using the finite element method (FEM). The validation concerned the fixed-free and fixed-pinned ends conditions of the beam. After this, the investigation was conducted by varying both the spring stiffness value from zero to infinite and the abscissa a from 0 to L. Different values of the fundamental circular frequency ω1 are determined for describing the dynamic behavior of the current vibrating beam system. The plotted curves show the variations of the transverse vibration circular frequency ω1 depending on the beam intermediate support location and stiffness. Keywords Fundamental angular frequency · Cantilever beam vibration · Flexible support · Translational spring · FEM

1 Introduction The present study focuses on the modeling of a cantilever beam of length L, with an intermediate flexible support, under the assumptions of the Euler–Bernoulli beam theory. The presence of a translational spring [3, 7–9] makes the beam vibration behavior more difficult to be apprehended and analyzed. This situation was examined for maximizing the beam natural frequency as reported in [11]. The particular case of a spring localization at the right end of the cantilever beam is indicated in [6]. Several investigations concerning the existence of an intermediate support with varying abscissa are also reported in [1, 2, 4, 10]. The resulting dynamic behaviors L. Chalah-Rezgui (B) · F. Chalah · S. E. Djellab · A. Nechnech Faculty of Civil Engineering, Usthb, 16111 Algiers, Algeria A. Bali Ecole Nationale Polytechnique, Algiers, Algeria © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_12

127

128

L. Chalah-Rezgui et al. ρ.A KT

EI

L

Fig. 1 Representation of a cantilever uniform beam of length L and elastically restrained in translation at an abscissa. EI: product of the material Young’s modulus times the beam moment inertia; ρ: density of the beam material; A: cross-sectional area of the beam; L: beam length; a: location of the spring; K T : stiffness of the spring

of the studied beam differ from classical beam studies by considering the flexibility of the vertical support. The stiffness of the vertical spring expresses different natures of the support. By affecting a value to the spring stiffness and fixing its position as presented in Fig. 1, a beam study is undertaken and the value of the circular frequency ω1 is sought. The investigation focuses first on the spring located at the right end of the cantilever beam and then at variable abscissas.

2 Spring Located at the Right Beam End Increasing the spring stiffness from zero to infinite changes the beam behavior from the initially cantilever beam to a fixed-elastically supported beam and then to a classic situation of a fixed-pinned beam, see Fig. 2.

KT

KT = 0

(b) Fixed-partially supported

(a) Fixed-Free

KT = (c) Fixed-Pinned

Fig. 2 Representations of a beam fixed at one end with a translation spring located at the second end Table 1 Values of α 2 = ω1 · (ρ · A · L 4 /E I )1/2 for extreme values of K *T when the spring is located at the right end Case study

K *T

α 2 theoretical values [5]

Fixed-free

0

1.87512 = 3.5160

Fixed-pinned

∞

3.92662

= 15.4182

α 2 present study 3.5160 15.4182

Fundamental Transverse Vibration Circular Frequency …

129

The comparisons concerned the particular situations where the translation spring stiffness is either K T = 0 (Fig. 2a) or K T = ∞ (Fig. 2c) for a = L. Table 1 summarizes the resulting values of dimensionless term α 2 = ω1 · (ρ · A · L 4 /E I )1/2 with ω1 transverse fundamental vibration circular frequency. For this purpose, a dimensionless variable K *T is introduced as written in Eq. 1: K T∗ = K T · L 3 /E I

(1)

As shown, the results are in good agreement. The case where the spring is located at the right end Fig. 2b, is treated to show its influence on the beam behavior. Figure 3 shows the variations of ω1 ·(ρ · A· L 4 /E I )1/2 as a function of K *T in the case where the spring is located at the beam right end. The K *T values smaller than unity (1) do not affect the dimensionless parameter characterizing the fundamental vibration circular frequency ω1 · (ρ · A · L 4 /E I )1/2 of the studied cantilever beam. The ω1 · (ρ · A · L 4 /E I )1/2 value increases progressively with the values of K *T in the interval 1–1000. Beyond the latter, the influence of the spring stiffness diminishes clearly and the curve becomes horizontal. A limit value of 15.4182 (theoretical) is reached for ω1 · (ρ · A · L 4 /E I )1/2 . This situation makes the analyzed uniform beam as fixed-pinned. 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1E-4

KT

1E-3

0,01

0,1

1

10

KT

100

1000

10000 100000

*

Fig. 3 ω1 · (ρ · A · L 4 /E I )1/2 variations as a function of K *T (K *T = K T .L 3 /E.I) in the case where the spring abscissa is L (a = L)

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3 Spring Located at a Variable Abscissa a The study is continued and conducted in order to show the influence of both the a/L ratio and K *T parameter characterizing the translational spring stiffness on the transverse vibrations of the studied cantilever uniform beam such restrained. By varying the locations of the spring from zero (0) to L and the K *T value from 0 to infinite (∞), the values of ω1 · (ρ · A · L 4 /E I )1/2 are obtained for each set of a/L ratio and K *T . The results are summarized on the plotted curves of Fig. 4. The resulting curves of Fig. 4 are quasi-linear for little values of K *T and become curvilinear by increasing K *T . Then, curved shapes appear beyond K *T ≈ 20 showing progressive maximal values of ω1 · (ρ · A · L 4 /E I )1/2 . This appears clearly for an overhang with a length (L−a) increasing from about 0.10L to less than about 0.23L particularly for high values of K *T . This is clearly observed for K *T = ∞. This situation corresponds to the second mode vibration of a uniform cantilever beam with ω1 · (ρ · A · L 4 /E I )1/2 = 22.034 (exact value given in the scientific literature) where the shape amplitude is null at the abscissa a = 0.774L.

24

*

KT =0

.A

E.

a=0.774L

KT=

*

KT =0.5

22

*

KT =2

20

.A

E.I

*

KT =5 *

KT =8

18

.A

E.I

a=L

KT=

*

KT =20

16

a

*

KT =30

KT

*

14

KT =40

L

*

KT =50

12

*

KT =70 *

10

KT =100 *

KT =150

8

*

KT =200 *

6

KT =1.d16

4 2 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

a/L Fig. 4 Variations of ω1 · (ρ · A · L 4 /E I )1/2 as a function of the a/L ratio and K *T (K T .L 3 /E.I) parameter

Fundamental Transverse Vibration Circular Frequency …

131

4 Conclusion The main contribution of this work is to study the effect of a translational spring located at a variable abscissa a on the fundamental vibration angular frequency of a uniform cantilever beam of length L. The preliminary validation concerned the values ω1 · (ρ · A · L 4 /E I )1/2 for two extreme values of K T (represented by K *T = 0 or K *T = ∞, respectively) located at a = L corresponding to the beam classical ends conditions fixed-free and fixed-pinned, respectively. Then, a curve giving the angular vibration frequency ω1 as a function of the spring stiffness (located at a = L) is plotted. After this, the location of the spring takes variable positions a and its stiffness is varied to show their effects on ω1 · (ρ · A · L 4 /E I )1/2 . The final results are represented by plotted curves showing the fundamental angular frequency in terms of the dimensionless value ω1 · (ρ · A · L 4 /E I )1/2 as a function of the (a/L) ratio and the parameter K *T characterizing the stiffness of the translational spring serving as an intermediate elastic support.

References 1. Chalah-Rezgui, L., Chalah, F., Djellab, S.E., Nechnech, A., Bali, A.: Free vibration of a beam having a rotational restraint at one pinned end and a support of variable Abscissa. In: Öchsner, A., Altenbach, H. (eds.) Mechanical and Materials Engineering of Modern Structure and Component Design, pp. 393–400. Springer International Publishing, Cham (2015). https://doi.org/ 10.1007/978-3-319-19443-1_32 2. Chalah-Rezgui, L., Chalah, F., Falek, K., Bali, A., Nechnech, A.: Transverse vibration analysis of uniform beams under various ends restraints. APCBEE Procedia 9, 328–333 (2014). https:// doi.org/10.1016/j.apcbee.2014.01.058 3. Darabi, M.A., Kazemirad, S., Ghayesh, M.H.: Free vibrations of beam-mass-spring systems: analytical analysis with numerical confirmation. Acta. Mech. Sin. 28, 468–481 (2012). https:// doi.org/10.1007/s10409-012-0010-1 4. Falek, K., Lila, R., Farid, C., Bali, A., Nechnech, A.: Structural element vibration analysis. Presented at the ICA 2013 Montreal, Montreal, Canada, pp. 065072–065072 (2013). https:// doi.org/10.1121/1.4799552 5. Gonçalves, P.J.P., Brennan, M.J., Elliott, S.J.: Numerical evaluation of high-order modes of vibration in uniform Euler-Bernoulli beams. J. Sound Vib. 301, 1035–1039 (2007). https://doi. org/10.1016/j.jsv.2006.10.012 6. Jafari, M., Djojodihardjo, H., Ahmad, K.A.: Vibration analysis of a cantilevered beam with spring loading at the tip as a generic elastic structure. Appl. Mech. Mater. 629, 407–413 (2014). https://doi.org/10.4028/www.scientific.net/AMM.629.407 7. Maurizi, M.J., Belles, P.: Vibrations of a beam clamped at one end and carrying a guided mass with an elastic support at the other. J. Sound Vib. 129, 345–349 (1989). https://doi.org/10.1016/ 0022-460X(89)90588-9 8. Maurizi, M.J., Bellés, P.M., Mart´ın, H.D.: An almost semicentennial formula for a simple approximation of the natural frequencies of Bernoulli-Euler beams. J. Sound Vib. 260, 191–194 (2003). https://doi.org/10.1016/S0022-460X(02)01129-X

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9. Maurizi, M.J., Rossi, R.E., Reyes, J.A.: Vibration frequencies for a uniform beam with one end spring-hinged and subjected to a translational restraint at the other end. J. Sound Vib. 48, 565–568 (1976). https://doi.org/10.1016/0022-460X(76)90559-9 10. Murphy, J.F.: Transverse vibration of a simply supported beam with symmetric overhang of arbitrary length. J. Test. Eval., JTEVA 25, 522–524 (1997) 11. Wang, D., Friswell, M.I., Lei, Y.: Maximizing the natural frequency of a beam with an intermediate elastic support. J. Sound Vib. 291, 1229–1238 (2006). https://doi.org/10.1016/j.jsv.2005. 06.028

Axial Fundamental Vibration Frequency of a Tapered Rod with a Linear Cross-Sectional Area Variation Farid Chalah, Lila Chalah-Rezgui, Salah Eddine Djellab and Abderrahim Bali

Abstract The axial vibration frequency of a tapered rod is investigated. The method of Rayleigh is used for this purpose. The taper type is relative to the linear variation of the cross section. The objective of the present investigation is to express the fundamental axial angular vibration frequency ω1 by a closed-form equation that takes account of the taper degree. For this purpose, the first mode shape function of the axial vibration of a uniform rod is adopted in the present investigation for a simplification aim. The necessary validation is made relatively to uniform and conical rods given in the scientific literature. Two formulas, depending on which end (large or small base) is fixed, are proposed for the fundamental angular frequency vibration of the tapered rod with a linear cross-sectional area variation. They are expressed as a function of the taper degree, Young’s modulus E, material density ρ, and the rod length L. Keywords Tapered rod · Axial vibration · Uniform rod · Rayleigh method · FEM

1 Introduction The vibration of the uniform rod was already studied in the literature [8]. The case where one end is free and the second is restrained by a translational spring was also investigated [3, 9]. On the other hand, the cross-sectional variation case was largely examined [4–7] using different approaches. The objective of the present work is to formulate a relationship for obtaining easily the fundamental axial angular vibration frequency ω1 of a tapered rod with a linear cross-sectional area variation as shown in Fig. 1. The considered approach is based on the energetic Rayleigh method where the posed problem is relative to expressing the shape function to be used for evaluating ω1 . F. Chalah (B) · L. Chalah-Rezgui · S. E. Djellab Faculty of Civil Engineering, Usthb, 16111 Algiers, Algeria A. Bali Ecole Nationale Polytechnique, Algiers, Algeria © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_13

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Small base

Lb

W

W

Lb

L Large base LB 3D

LB view

Front view

Side view

Fig. 1 Representation of a fixed-free tapered rod with a linear cross-sectional area variation. L B : length of a large base; L b : length of the small base; W: width of the side face; L: tapered rod length

The Rayleigh method also called the energetic method of Rayleigh is widely used for analyzing a mechanical system vibrating harmonically. The considered formula is adapted for an axial movement, as given after in Eq. 1.  ω1,R =

E.A(x).(ψ  (x))2 dx  m(x).(ψ(x))2 dx

(1)

where E A(x) m(x) ψ(x)

Young’s modulus. cross-sectional area variation. mass variation. shape function.

2 Analysis of a Tapered Rod with a Linear Cross-Sectional Area Variation In the Oxy reference, a tapered rod is considered in Fig. 2, consisting of a material with the elastic modulus E and cross-sectional area A(x) at the x abscissa subjected to a non-uniform axial loading N(x). The dx element equilibrium is governed by Eq. 2: Fig. 2 Representation of a tapered rod with a linear variable cross-sectional area loaded by an axial force N(x)

y A0 O

x

A(x) dx L

N(x) x

Axial Fundamental Vibration Frequency of a Tapered Rod …



135

 E.A(x).u  (x) = N (x)

(2)

The development of the previous equation leads to:   E. A (x).u  (x) + A(x).u  (x) = N (x)

(3)

The present problem is resolved by considering that the variation of the crosssectional area, as shown in Fig. 3, is in the form of Eq. 4: A(x) = A0 .(x/a)

(4)

where x abscissa of the current cross section; see Fig. 3. A0 cross-sectional value of the tapered rod at its small base corresponding to the abscissa a. a length separating the position of the cross-sectional area A0 and the intersection resulting after the extension of the two lateral sides of the tapered rod. The application of the Rayleigh formula for evaluating the fundamental axial circular vibration frequency requires a shape function ψ(x). This required function must satisfy the end conditions for allowing an accurate evaluation of ω1 . The classical proposition commonly admitted and applied in analyzing the transverse vibration of a harmonically oscillating system consists of using a particular shape function ψ(x). It results from the normalization of the displacement function obtained by the application of the loading due to the gravitational forces (self-weight) acting in the direction of the movement. This approach is adapted here for the analysis of the axial vibration of a non-uniform rod. A difficulty of evaluating this expression appears because the shape function ψ(x) depends on the length a introduced before. Therefore, the shape function ψ u (x) of a uniform rod will be used to evaluate the whole expression in the case of non-uniform rod. Then, Eq. 1 is rewritten in the form of Eq. 5.  ω1,R =

E.A(x).(ψu (x))2 dx  m(x).(ψu (x))2 dx

(5)

where ψ u (x) shape function of a uniform rod. Fig. 3 Representation of a a bar with variable cross-sectional area in the form of A(x) = A0 .(x/a)

y

x

A(x)

A0

O

a

dx L

N(x)

x

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The mass variation follows that of the cross-sectional area of Eq. 4, as follows: m(x) = ρ.A(x)

(6)

where ρ density of the material.

3 Vibration of Uniform Rod Two methods, the Rayleigh method and the finite element method (FEM) respectively (for evaluating the fundamental angular vibration frequency ω1 of a vibrating uniform rod), are exposed in Sects. 3.1 and 3.2.

3.1 Vibration of Uniform Rod Using the Rayleigh Method In order to obtain the shape function to be utilized in the Rayleigh formula, the governing differential equation for the axial displacement of a uniform rod, as illustrated in Fig. 4, is: E.A.u  (x) = ρ.A.g

(7)

where g gravity acceleration in m/s2 . A uniform cross-sectional area. ρ.A.g uniformly distributed loading (self-weight) acting horizontally. Figure 4 represents a uniform rod subjected to an axial loading corresponding to a uniformly distributed loading ρ.A.g (self-weight) applied horizontally. The solution of Eq. 7 is: u(x) =

Fig. 4 Representation of a uniform rod loaded by its self-weight

x 2 − 2L .x ρ.g . 2 E

(8)

y

x

A(x)=A N(x)= ρ.A.g x

O L

E, ρ

Axial Fundamental Vibration Frequency of a Tapered Rod …

137

with u(0) = 0 and u (L) = 0. The first mode shape function to be considered, to find ω1 , consists of normalizing the above axial displacement function. It is obtained by dividing this function by the maximal axial displacement at the free end, as expressed in Eq. 9: −x 2 + 2L .x u(x) = u(L) L2

ψu (x) =

(9)

By considering Eq. 6, replacing A(x) by A corresponding to a constant cross section in Eq. 5 and after simplifying it (term A) in both the numerator and denominator, the fundamental angular vibration frequency ω1,RCu of a uniform rod is expressed by:  ω1,RCu =

  E. (ψCu (x))2 dx  ρ. (ψCu (x))2 dx

(10)

The given Eqs. 11–13 are developed for the evaluation of ω1,RCu : 1 4 (x − 4L .x 3 + 4L 2 x 2 ) L4

(ψCu (x))2 =

ψu (x) =

1 (−2x + 2L) L2

  2 1 ψu (x) = 4 (4x 2 − 8L .x + 4L 2 ) L

(11) (12) (13)

Integrating along with the interval [0, L], the previous Eqs. 11 and 13 will give ω1,RCu for the following expression:  ω1,RCu =

E.(4/3L 3 − 4L 3 + 4L 3 ) ρ.(1/5L 5 − 4/4L 5 + 4/3L 5 )

(14)

This after few manipulations is replaced by:  1 E.(4/3) ρ.(3 − 15 + 20)/15 L 2

ω1,RCu =

(15)

or  ω1,RCu =

5.4 E = 8 ρ.L 2



 5 E . 2 ρ.L 2

(16)

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The final result is  ω1,RCu = 1.58114.

E ρ.L 2

(17)

A formulation of ω1,RC as a function of the mass m = ρ.A will give:  ω1,RCu = 1.58114.

 E.A E.A = 1.58114. ρ.A.L 2 m.L 2

(18)

The theoretical fundamental angular vibration frequency of a uniform rod as shown in Fig. 4 is: ω1,Theo.

π = 2



 E E = 1.57080. ρ.L 2 ρ.L 2

(19)

The error of evaluating ω1,RCu is less than 0.7%.

3.2 Axial Vibration of a Uniform Rod Using the FEM Analysis In this section, the fundamental vibration angular frequency in the case of a uniform rod is evaluated by the FEM analysis by considering the elementary stiffness matrix of Eq. 20:   E e .Ae 1 −1 [K e ] = −1 1 Le

(20)

While that of the mass is expressed in two forms: The first one consists of dividing by two the mass of the elementary rod and affects each resulting portion to each end as follows:   1/2 0 (21) [Me ] = ρe .Ae .L e 0 1/2 And the second is issued from a FEM classical formulation, as given in Eq. 22:  [Me ] = ρe .Ae .L e

1/3 1/6 1/6 1/3

 (22)

Axial Fundamental Vibration Frequency of a Tapered Rod … Table 1 ω0,FEM dimensionless parameter values according to the two mass matrices forms

139

Elements

Form 1 of [M e ]

Form 2 of [M e ]

1

1.41421

1.73205

2

1.53073

1.61142

4

1.56072

1.58091

8

1.56827

1.57332

16

1.57017

1.57143

32

1.57064

1.57095

64

1.57076

1.57084

128

1.57079

1.57081

256

1.57079

1.57080

512

1.57080

1.57080

Note The exact value of the ω0 dimensionless parameter is π /2 corresponding to about 3.14159/2 = 1.57080. This result is reached for 16 elements for the two forms of the mass matrices with 5 digits. The errors are about 0.04% and 0.01 for 16 and 32 elements, respectively. The accuracy increases greatly beyond 32 elements

The values of ω1 will be functioned of the ω0,FEM dimensionless parameter in the formula:  E (23) ω1,FEM = ω0,FEM . ρ.L 2 Table 1 summarizes the results of the ω0,FEM dimensionless parameter values to consider in the expression evaluating ω1,FEM in the case of a uniform rod vibrating axially. The considered number of uniform elementary rods is in the sequence of numbers 1, 2, 4, 8, 16, 32, 64, 128, 256, and 512. The calculation is carried out according to the two mass matrices forms. Figure 5 shows the representation of a fixed-free uniform rod divided into 2, 4, 8, and 16 elementary rods. Fig. 5 Representation of a fixed-free uniform rod discretized by 2, 4, 8, and 16 elementary uniform rods

N=2 N=4 N=8 N=16

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4 Fundamental Vibration Angular Frequency of a Tapered Rod with a Linear Cross-Sectional Area Variation The shape function of Eq. 9 corresponding to the axial vibration of a uniform rod as shown in Fig. 4 will be used for analyzing the tapered rod by the Rayleigh formula in the case of a linear cross-sectional area variation, such as given in Fig. 6. The expression of the linear cross-sectional area variation is as expressed in Eq. 4, and that of the mass is defined in Eq. 6.

4.1 Analysis of a Linear Tapered Rod Fixed at Its Large Base For the origin of the axis located at the large base, the variation of the cross section A(x) is: A(x) = A0 .(a + L − x)/a

(24)

The product A(x) times (ψ  Cu )2 will give: 2   (x) = A(x). ψCu



A0 .(4ax 2 − 8a.L x + 4a L 2 + 4L x 2 a.L 4

− 8L 2 x + 4L 3 + −4x 3 + 8L x 2 − 4L 2 .x)

(25)

The integration of Eq. 25 along with the interval 0 to L gives:

Fig. 6 Representations of a tapered rod with a linear distributed mass m(x) = ρ.A(x). a Large base fixed and b small base fixed

y

(a) A0

x

A(x)

x O

a

L

(b) y

x

A0

A(x)

x

O a

L

Axial Fundamental Vibration Frequency of a Tapered Rod …



141



A0 4 3 8 4 4 8 4 4 4 8 4 4 4 3 3 4 . − + 4a L + − + 4L − + − a L a.L L L L L L a.L 4 3 2 3 2 4 3 2

(26)

After simplification, it is rewritten in Eq. 27:



A0 4 12 . a+ L − 3L a.L 3 3

(27)

It results in:



A0 4 . a+L a.L 3

(28)

The term (ψ Cu )2 is expressed in Eq. 29:

−x 2 + 2L .x L2

2 =

1 4 (x − 4L .x 3 + 4L 2 x 2 ) L4

(29)

The product A(x).(ψ Cu )2 is given in Eq. 30:

A0 .(ax 4 − 4a.L x 3 + 4a L 2 x 2 + L x 4 − 4L 2 x 3 + 4L 3 x 2 − x 5 + 4L .x 4 − 4L 2 x 3 ) (30) a.L 4

The integration of Eq. 30 along the interval [0, L] is expressed in the terms of Eqs. 31–35:

A0 1 1 1 1 1 1 . a L 5 − 4a.L L 4 + 4a L 2 L 3 + L L 5 − 4L 2 L 4 + 4L 3 L 3 a.L 4 5 4 3 5 4 3

1 1 1 (31) − L 6 + 4L .L 5 − 4L 2 L 4 5 4 6

1 A0 1 1 1 1 1 1 1 . a L − a.L + 4a L + L 2 − 4L 2 + 4L 2 − L 2 + 4.L 2 − 4L 2 (32) a

5



3

5

4

3

6

5

4







A0 1 1 4 4 1 4 . aL −1+ + L2 −1+ − + −1 a 5 3 5 3 6 5

A0 .(a L(6 − 30 + 40) + L 2 (6 − 60 + 40 − 5 + 24)) 30a

A0 .(16a L + 5L 2 ) 30a

(33) (34) (35)

At the end, the evaluation of the fundamental vibration angular frequency of a tapered rod having a linear cross-sectional area variation for which the large base is fixed becomes:

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ω1,RCT

 =

 A0  .(4/3a + L) E. a.L  A0  ρ. 30.a .(16a.L + 5.L 2 )

(36)

Simplifying the fraction A0 /a in both the numerator and the denominator and rearranging the remaining terms gives:  ω1,RCT =

 1  .(4a + 3L) E. 3.L L ρ. 30 .(16a. + 5.L)

(37)

The final resulting expression of ω1,RCT is:  ω1,RCT =

  √ 10.(4a + 3L) E (4a + 3L) E = 10 (16a. + 5.L) ρ.L 2 (16a. + 5.L) ρ.L 2

(38)

A second version of this formula is given by considering the ratio α = a/L: ω1,RCT =

√



(4α + 3) 10 (16α + 5)

 E ρ.L 2

(39)

4.2 Analysis of a Linear Tapered Rod Fixed at Its Small Base Similar developments of the whole expressions produced previously are considered for expressing the fundamental angular axial vibration frequency of a tapered rod, with a linear variation of the cross-sectional area, which is fixed at its small base. The shape to be used is in the form of Eq. 40: ψCu (x) =

1 (−(x − a)2 + 2L .(x − a)) L2

(40)

1 (−2.x + 2a + 2L) L2

(41)

The term ψ  Cu gives:  (x) = ψCu

Thus (ψ iCu )2 is equal to: 2   1 ψCu (x) = 4 (−2x + 2a + 2L)2 L

(42)

Axial Fundamental Vibration Frequency of a Tapered Rod …

143

It is developed into: 2   1 ψCu (x) = 4 (4x 2 − 4a.x − 4L .x + 4a 2 − 4a.x + 4.a.L L − 4L .x + 4a.L + 4L 2 )

(43)

And rewritten in:   2 1 ψCu (x) = 4 (4x 2 − 8a.x − 8L .x + 4a 2 + 8.a.L + 4L 2 ) L

(44)

Carrying out the multiplication of A(x) by (ψ  Cu )2 leads to: x  1 2   . 4 (4x 2 − 8a.x − 8L .x + 4a 2 + 8.a.L + 4L 2 ) (45) (x) = A0 A(x). ψCu a

L   2 A(x). ψCu (x) =

A0 (4x 3 − 8a.x 2 − 8L .x 2 + 4a 2 x + 8.a.L x + 4L 2 .x) a.L 4

(46)

Integrating in the interval a to a + L the expression of Eq. 46 gives: a+L 

  2 A(x). ψCu (x) dx =



A0 8 .(((a + L)4 − a 4 ) − a.((a + L)3 − a 3 ) 4 a.L 3

a

8 L .((a + L)3 − a 3 ) + 2a 2 ((a + L)2 3 − a 2 ) + 4a L((a + L)2 − a 2 ) + 2L 2 ((a + L)2 − a 2 ))

−

(47)

After expressing the terms between brackets, it is rewritten in Eqs. 48–50: a+L 

  2 A(x). ψCu (x) dx =



A0 8 .(((a + L)4 − a 4 ) − a.((a + L)3 a.L 4 3

a

8 L .((a + L)3 − a 3 ) + 2a 2 ((a + L)2 − a 2 ) 3 + 4a L((a + L)2 − a 2 ) + 2L 2 ((a + L)2 − a 2 )) − a3) −

a+L 

  2 A(x). ψCu (x) dx =



(48)

A0 .((4a 3 L + 6a 2 L 2 + 4a L 3 + L 4 ) a.L 4

a

8 8 − a.(3a 2 L + 3a L 2 + L 3 ) − L .(3a 2 L + 3a L 2 + L 3 ) 3 3 + 2a 2 (2a L + L 2 ) + 4a L(2a L + L 2 ) + 2L 2 (2a L + L 2 ))

(49)

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F. Chalah et al. a+L 



 A0 . 4a 3 L(4 − 8 + 4) + a 2 L 2 (6 − 8 − 8 + 2 + 8) a.L 4



8 8 (50) +a L 3 4 − − 8 + 4 + 4 + L 4 1 − + 2 3 3

  2 A(x). ψCu (x) dx =

a

The simplification of Eq. 50 leads to: a+L 

  2 A(x). ψCu (x) dx =





4 3 1 4 A0 . a L L + a.L 4 3 3

(51)

A0 .(4a + L) 3a.L

(52)

a

That results in: a+L 

  2 A(x). ψCu (x) dx =



a

The term (ψ 1Cu )2 is expressed in Eqs. 53–56: 1 (−(x − a)2 + 2L .(x − a))2 L4

(53)

1 (−x 2 + 2ax + 2L x − a 2 − 2a L)2 L4

(54)

(ψCu (x))2 = (ψCu (x))2 =

1 4 (x − 2ax 3 − 2L x 3 + a 2 x 2 + 2a L x 2 − 2ax 3 L4 + 4a 2 x 2 + 4a L x 2 − 2a 3 x − 4a 2 L x − 2L x 3 + 4a L x 2 + 4L 2 x 2

(ψCu (x))2 =

− 2a 2 L x − 4a L 2 x + a 2 x 2 − 2a 3 x − 2a 2 L x + a 4 + 2a 3 L + 2a L x 2 − 4a 2 L x − 4a L 2 x + 2a 3 L + 4a 2 L 2 )

1 4 (x − 4ax 3 − 4L x 3 + 6a 2 x 2 + 12a L x 2 + 4L 2 x 2 L4 − 4a 3 x − 12a 2 L x − 8a L 2 x + 4a 3 L + 4a 2 L 2 + a 4 )

(55)

(ψCu (x))2 =

(56)

The product A(x).(ψ Cu )2 is written as follows: A(x).(ψCu (x))2 =

A0 a L4



1 5 (x − 4ax 4 − 4L x 4 + 6a 2 x 3 L4

+ 12a L x 3 + 4L 2 x 3 − 4a 3 x 2 − 12a 2 L x 2 − 8a L 2 x 2 + 4a 3 L x + 4a 2 L 2 x + a 4 x)

(57)

The integration of the previous equation in the interval [a, a + L] results in the expressions of Eqs. 58 and 59:

Axial Fundamental Vibration Frequency of a Tapered Rod …

145

1 1 4 ((a + L)6 − a 6 ) − a((a + L)5 L4 6 5 4 6 12 − a 5 ) − L((a + L)5 − a 5 ) + a 2 ((a + L)4 − a 4 ) + a L((a 5 4 4 4 4 + L)4 − a 4 ) + L 2 ((a + L)4 − a 4 ) − a 3 ((a + L)3 − a 3 ) 4 3 12 2 8 2 4 3 3 − a L((a + L) − a ) − a L ((a + L)3 − a 3 ) + a 3 L((a + L)2 3 3 2

4 1 −a 2 ) + a 2 L 2 ((a + L)2 − a 2 ) + a 4 ((a + L)2 − a 2 ) (58) 2 2

A0 1 A(x).(ψCu (x))2 = (1/6(6a L 5 + 15a 2 L 4 + 20a 3 L 3 a L4 L4

A(x).(ψCu (x))2 =

A0 a L4



+ 15a 4 L 2 + 6a 5 L + L 6 ) − 4/5a(5a L 4 + 10a 2 L 3 + 10a 3 L 2 + 5a 4 L + L 5 ) − 4/5L(5a L 4 + 10a 2 L 3 + 10a 3 L 2 + 5a 4 L + L 5 ) + 6/4a 2 (4a L 3 + 6a 2 L 2 + 4a 3 L + L 4 ) + 3a L(4a L 3 + 6a 2 L 2 + 4a 3 L + L 4 ) + L 2 (4a L 3 + 6a 2 L 2 + 4a 3 L + L 4 ) − 4/3a 3 (3a L 2 + 3a 2 L + L 3 ) − 12/3a 2 L(3a L 2 + 3a 2 L + L 3 ) − 8/3a L 2 (3a L 2 + 3a 2 L + L 3 ) + 2a 3 L(2a L + L 2 ) + 2a 2 L 2 (2a L + L 2 ) + 1/2a 4 (2a L + L 3 ))

(59)

The simplification of Eq. 59 is expressed in the terms of Eq. 60:



A0 A0 .L 6 5 .(11/30.L .(11.L + 16a) + 8/15a L ) = a.L 4 30a

(60)

Thus, the fundamental vibration angular frequency of a tapered rod with a linear cross-area variation in the case where its small base is fixed is:    E. A0 (4a + L) 3a.L

ω1,RCT =  0 .L  (61) ρ. A30a (11.L + 16a) After simplifying the term (A0 /3a), it becomes: ω1,RCT =

√

 10

E.(4a + L) ρ.L 2 (16.a + 11.L)

(62)

A second version of Eq. 62 is obtained by considering α = a/L: ω1,RCT (α) =

√



 (4α + 1) E . 10 (16.α + 11) ρ.L 2

(63)

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Remark For high values of α, the tapered rod will take a uniform form and then the ω1,RCT value will tend to: ω1,RCT(αhigh )

√ = 10



    4 E 5 E E . . = = 1.58114. 16 ρ.L 2 2 ρ.L 2 ρ.L 2

(64)

This concerns the two situations of ends restraints conditions regardless of which one of the two ends is fixed while the other is free. The taper degree is reduced, and therefore, ω1,RCT(αhigh ) takes the value of 1.58114 which is equal to the constant of Eq. 17.

4.3 Variations of ω1,RCT .(ρ.L2 /E)1/2 of a Tapered Rod Fixed at Its Large Base The results obtained by Eq. 39 are superposed as illustrated in Fig. 7 on the digitalized curve which is given in an existing research work in the scientific literature [4]. As shown, the curve given in Fig. 7 follows the same pattern as that of the plotted points. 2,5 2,4 2,3

ω1.(ρL2/E)1/2

2,2 2,1 2,0 1,9

O O

1,8 1,7

O

1,6 1,5 1E-3

0,01

0,1

1

10

100

a/L Fig. 7 Variations of the ω1,RC .(ρ.L 2 /E)1/2 dimensionless parameter obtained by the suggested formula and [4] for a tapered rod fixed at its large base, with a linear cross-area variation

Axial Fundamental Vibration Frequency of a Tapered Rod …

147

1,6

O

1,5

ω1.(ρL2/Ε )1/2

1,4 1,3

O

O

1,2 1,1 1,0 0,9 1E-3

0,01

0,1

1

10

100

a/L Fig. 8 Variations of the ω1,RCT .(ρ.L 2 /E)1/2 dimensionless parameter obtained by the suggested second formula and [4] for a tapered rod fixed at its small base with a linear cross-area variation

4.4 Variations of ω1,RCT .(ρ.L2 /E)1/2 of a Tapered Rod Fixed at Its Small Base Similarly to Sect. 4.3, the curve plotted using Eq. 63 is superposed in Fig. 8 with a second digitalized curve for its corresponding restraints conditions given in [4]. As shown, the two resulting curves are comparable and this, as hoped, facilitates the evaluation of the fundamental vibration angular frequency of such tapered rod.

5 Variations of ω1 of a Tapered Rod Using the FEM The considered elementary stiffness and mass matrices of the tapered rod represented in Fig. 9 are obtained by the cross-sectional area located at the middle height of each elementary tapered rod.

5.1 Case Where the Large Base Is Fixed Figure 10a shows the variations of the ω1,FEM .(ρ.L 2 /E)1/2 dimensionless parameter curves in the function of a/L obtained by using 2, 4, 8, 16, and 64 elementary uniform rods. The upper curve is then superposed in Fig. 10b to the plotted

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Fig. 9 Discretization of a tapered rod in two (02) elementary uniform rods

L/2

L/4 L/4

(a)

2,5 2,4 2,3 2 rods 4 rods 8 rods 16 rods 64 rods

2,2

ω1.(ρL2/Ε)1/2

Fig. 10 Variations of ω1,FEM .(ρ.L 2 /E)1/2 dimensionless parameter values. a ω1,FEM .(ρ.L 2 /E)1/2 values with N number of rods as parameter, b Superposition of ω1,FEM .(ρ.L 2 /E)1/2 and ω1,Formula .(ρ.L 2 /E)1/2

2,1 2,0 1,9 1,8 1,7 1,6 1,5 1E-3

(b)

0,01

0,1

2,5

1

10

100

a/L

2,4

ω1.(ρL2/E)1/2

2,3 2,2

ω1,Formula.(ρL /E)

2,1

ω1,FEM.(ρL /E)

2

2

1/2

1/2

2,0 1,9 1,8 1,7 1,6 1,5 1E-3

0,01

0,1

1

10

100

a/L

curve ω1,RCT .(ρ.L 2 /E)1/2 based on the proposed formula. The calculation concerns the concentrated form of the elementary mass matrix. The left side of the plotted curve of Fig. 10a characterizes high degrees of taper where an increase in the number elementary rods is necessary to obtain a convergence to ω0,FEM .(ρ.L 2 /E)1/2 . As highlighted in Fig. 10b, the curve variation of the proposed formula is similar to that of ω0,FEM .(ρ.L 2 /E)1/2 . A number of 64 elementary uniform rods corresponding to the high size matrices are nevertheless necessary to give a

Axial Fundamental Vibration Frequency of a Tapered Rod …

149

good approximate of ω1,RC .(ρ.L 2 /E)1/2 for high taper degrees. But, on the right side of the curve as shown in Fig. 10b, the values are in good agreement with those of the FEM for which a number of 4 elementary rods are sufficient to obtain an accurate value. A research work concerning an application of a version of the Rayleigh method reported in reference [2] gives two values for the dimensionless angular frequency corresponding to 61/2 = 2.4495 and 2.4142, where the errors are about 1.86% and 0.39%, respectively, in comparison to the exact value. The first value is obtained with the same error of 0.01858 ≈ 1.86% by using the suggested closed-form formula by considering, for α = 0: √

 10

3 √ = 6 = 2.4495 5

(65)

and the second one is reached using the FEM. In parallel to this, other research works reported the value 2.4048 as being the exact value when studying linear vibratory systems [1] and vibration analysis of a tampered bar by differential transformation [10]. As shown, an evident advantage is offered by allowing the designer to evaluate a large broad of taper variations by using a simplified formula.

5.2 Case Where the Small Base Is Fixed Figure 11 shows the variations of the curves ω1,FEM .(ρ.L 2/ E)1/2 depending on a/L for 2, 4 elementary uniform rods. As illustrated in Fig. 11, the obtained value of ω1,RC .(ρ.L 2/ E)1/2 for very small values of the taper degree α using the Rayleigh method (when employing a uniform rod shape) is of the same order than the ω0,FEM .(ρ.L 2/ E)1/2 for 2 and/or 4 elementary rods. For high values of α, the use of a number of elementary rods beyond 4 is necessary to obtain a good correspondence with the curve issued from the formula for the case where the small base is fixed. The value of 1.58114 as expected is of the same accuracy than the exact value of 1.57080.

6 Conclusion The problem posed in analyzing the axial vibrations of a uniform rod was largely treated in the scientific literature. The present study focused on the case of a linear cross-sectional area variation. The plotted curve based on the proposed formula is close to the set of points obtained by the digitalization of a curve describing the variation of ω1 resulting from the Rayleigh method. This means that the application of the proposed formula gives accurate values for the fundamental axial circular

ω1.(ρL2/E)1/2

150

F. Chalah et al. 1,60 1,55 1,50 1,45 1,40 1,35 1,30 1,25 1,20 1,15 1,10 1,05 1,00 0,95 0,90 0,85 0,80 0,75 0,70 1E-3

ω 1,Formula ω 1,FEM using 2 rods ω 1,FEM using 4 rods

0,01

0,1

1

10

100

a/L Fig. 11 Variation of ω1 .(ρ.L 2 /E)1/2 dimensionless parameter values

frequency of an oscillating tapered rod for a linear cross-area variation. This is possible by utilizing the tapered rod dimensions features (a and L) expressed by the dimensionless variable α.

References 1. Bert, C.W.: Relationship between fundamental natural frequency and maximum static deflection for various linear vibratory systems. J. Sound Vib. 162, 547–557 (1993). https://doi.org/ 10.1006/jsvi.1993.1139 2. Bert, C.W.: Application of a version of the Rayleigh technique to problems of bars, beams, columns, membranes, and plates. J. Sound Vib. 119, 317–326 (1987). https://doi.org/10.1016/ 0022-460X(87)90457-3 3. Bhat, R.B.: Obtaining natural frequencies of elastic systems by using an improved strain energy formulation in the Rayleigh-Ritz method. J. Sound Vib. 93, 314–320 (1984). https://doi.org/ 10.1016/0022-460X(84)90316-X 4. Chalah, F., Djellab, S.E., Chalah-Rezgui, L., Falek, K., Bali, A.: Tapered beam axial vibration frequency: linear cross-area variation case. APCBEE Procedia 9, 323–327 (2014). https://doi. org/10.1016/j.apcbee.2014.01.057 5. De Rosa, M.A., Franciosi, C.: The optimized Rayleigh method and mathematicain vibrations and buckling problems. J. Sound Vib. 191, 795–808 (1996). https://doi.org/10.1006/jsvi.1996. 0156 6. Eisenberger, M.: Exact longitudinal vibration frequencies of a variable cross-section rod. Appl. Acoust. 34, 123–130 (1991). https://doi.org/10.1016/0003-682X(91)90027-C

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7. Friedman, Z., Kosmatka, J.B.: Exact stiffness matrix of a nonuniform beam—I. Extension, torsion, and bending of a bernoulli-euler beam. Comput. Struct. 42, 671–682 (1992). https:// doi.org/10.1016/0045-7949(92)90179-4 8. Ma, H.: Exact solutions of axial vibration problems of elastic bars. Int. J. Numer. Meth. Eng. 75, 241–252 (2008). https://doi.org/10.1002/nme.2254 9. Shangchow, C.: The fundamental frequency of an elastic system and an improved displacement function. J. Sound Vib. 81, 299–302 (1982). https://doi.org/10.1016/0022-460X(82)90211-5 10. Zeng, H., Bert, C.W.: Vibration analysis of a tapered bar by differential transformation. J. Sound Vib. 242, 737–739 (2001). https://doi.org/10.1006/jsvi.2000.3372

Vibration of an SDOF Representing a Rigid Beam Supported by Two Unequal Columns with One Mounted on a Flexible Base Lila Chalah-Rezgui, Farid Chalah, Salah Eddine Djellab, Ammar Nechnech and Abderrahim Bali

Abstract The structural dynamic analysis considers a basic mechanical vibrating system to represent a single degree of freedom. Such a system commonly formed by an infinitely rigid beam supported by two columns is used to establish the dynamic equilibrium equation. The analyzed single degree of freedom noted 1-DOF in this contribution considers different heights for the columns while a torsion spring is disposed at the first column base. Thus, the lateral stiffness is constituted by the sum of the contributions of the first column, the torsional spring, and the second column. This study allowed the determination of the vibration period of the 1-DOF by using the finite element method for various heights ratios and different values of the torsion spring stiffness. The findings of the conducted investigations are presented on plotted curves giving the vibration period T as a function of both the heights ratio and the torsion spring stiffness value. Keywords Frame · One story · Torsion spring · Column stiffness · Unequal heights

1 Introduction Many dynamic situations may be represented by two built-in columns with the same constant height h supporting a rigid beam of a total mass m. The mass of the vibrating system is concentrated at the floor and the supporting columns are considered mass less and inextensible in the vertical axis. This model, as shown in Fig. 1, is usually used in the scientific literature [3, 6] to express the dynamic equilibrium equation of an oscillating system of a single degree of freedom noted 1-DOF. The columns at a same level acting in parallel to resist against a lateral load are represented by two columns. Thus, the total level stiffness corresponds to the addition of that of each column. L. Chalah-Rezgui (B) · F. Chalah · S. E. Djellab · A. Nechnech Faculty of Civil Engineering, Usthb, 16111 Algiers, Algeria A. Bali Ecole Nationale Polytechnique, Algiers, Algeria © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_14

153

154

L. Chalah-Rezgui et al.

Fig. 1 Representation of a 1-DOF formed by a rigid beam supported by two columns of equal length h

Infinitely rigid beam Beam mass :mb h E.I

E.I

The rigid beam ensures a same lateral displacement of the top of the two columns. This permits to add linearly the contributions of the lateral stiffness of each column evaluated separately to obtain the total stiffness of the 1-DOF. After these considerations, the lateral vibration period T of the system is calculated. The classical treated examples in the scientific literature deal generally with two parallel columns with a same height h where the ends conditions for the two columns may be pinned-fixed or fixed-fixed. The stiffness of a given column depending on its ends conditions is: kc = 3E.I / h 3 : for a pinned-fixed column (F-P) and kc = 12E.I / h 3 : for a fixed-fixed column(F-F). where I moment of inertia about the neutral axis of the column, h column height, E Young’s modulus of the column material. Thus, the total stiffness of the global system due to the contribution of these two columns depending on their ends conditions may be: 6EI/h3 , 15EI/h3 , or 24EI/h3 . In the present study, a one story vibrating system (1-DOF) constituted by a rigid beam supported by two columns with unequal heights is investigated. The lower node of the first column is considered spring-hinged while the base of the second column is first set pinned (Fig. 2a) and after fixed (Fig. 2b). The effect of adding a torsional spring with a stiffness K r makes the base of the left column (fixed at its upper part) initially pinned (K r = 0), partially fixed

(a)

(b) Mass: m E.I K=3.E.I/h23 h2 I

E.I h1

K=12.E.I/ h23

Kr

Fig. 2 Representations of a rigid beam supported by two unequal columns where the first one is spring-hinged at its lower end while the second column is a pinned-fixed, b fixed-fixed

Vibration of an SDOF Representing a Rigid Beam …

155

(K r = 0), or fixed (K r = ∞). Several research works were concerned by the vibrations of restrained beams [4, 5, 7]. Different methods exist to treat this kind of problems and some of them usually use the finite element method (FEM) [1, 2]. This study concerns the variations of both the ratio α = h1 /h2 and the torsional spring K r value for a constant product E.I. The results of the vibration period T are first presented for extreme K r values cases obtained by hand calculus and determinant search. Then plotted curves depending on α = h1 /h2 and the torsional spring K r as a parameter are given. The investigation is conducted by neglecting the shear strains.

2 Vibration Period for Extreme Values of K r The extreme values that the torsional spring stiffness K r may take are considered to validate the first comparisons. Thus, two study cases are considered depending on the second column base restraints, as shown in Fig. 3.

2.1 Right Column Pinned-Fixed The vibration periods T are evaluated for α = 1/3 or 10/3 assumed in this context as being practical extreme heights ratios. In the case where K r = 0 (pinned-fixed first column), they are, respectively:

(a)

Kr=0

Kr=

Kr=0

Kr=

(b)

Fig. 3 Study cases for extreme variations of K r for a given α = h1 /h2 ratio where the base of the second column is first set pinned (a) and after fixed (b)

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L. Chalah-Rezgui et al.

  3 m.h 32 m.h 32 = 0.6855 T = 2π + 3. . 0.33333 E.I E.I     3 m.h 32 m.h 32 T = 2π + 3. . = 3.5796 3.33333 E.I E.I 



(1)

(2)

For very high value of K r (fixed-fixed first column), the values of T are:   12 m.h 32 m.h 32 = 0.3474 + 3. . T = 2π 0.33333 E.I E.I      12 m.h 32 m.h 32 T = 2π = 3.4463 + 3. . 3 3.3333 E.I E.I 



(3)

(4)

As shown, Eqs. (1)–(4) give quickly the values of the dimensionless term T.(EI/mh32 )1/2 characterizing the vibration period. The analysis of the 1-DOF for small values of h1 (α ≈ 1/3) leads, as expected, to a behavior essentially governed by the first column only. This simplifies Eqs. (1) and (3), rewritten in Eqs. (5) and (6):   3 m.h 32 m.h 32 T = 2π . = 0.6980 0.33333 E.I E.I      12 m.h 32 m.h 32 = 0.3490 . T = 2π 0.33333 E.I E.I 



(5)

(6)

The results obtained by using the simplified Eqs. (5) and (6) are of the same order of accuracy comparatively to those found by the use of Eqs. (1) and (3). Equations (2) and (4) in the case of α = 10/3 are simplified to 2π/31/2 (mh32 /EI)1/2 = 3.6276 (mh32 /EI)1/2 . The T.(EI/mh32 )1/2 values obtained applying the formulas are in good agreement with those issued from the null determinant solving. Table 1 summarizes the results on values of T.(EI/mh32 )1/2 obtained in the case of a pinned-fixed right column for α = 1/3 and 10/3. The last column represents the results issued from the null determinant search. This latter is derived from the global stiffness and mass matrices.

Vibration of an SDOF Representing a Rigid Beam …

157

Table 1 Values T.(EI/mh32 )1/2 obtained in the case of a pinned-fixed right column for α = 1/3 and 10/3 Case

Column 1

α = h1 /h2

T.(EI/mh3 )1/2 hand calculus

T.(EI/mh3 )1/2 simplified calculus

T.(EI/mh3 )1/2 determinant

1

P-F

1/3

0.6855

0.6980

0.6856

2

P-F

10/3

3.5796

3.6276

3.5796

3

F-F

1/3

0.3474

0.3490

0.3475

4

F-F

10/3

3.4463

3.6276

3.4463

Table 2 Values T.(EI/mh32 )1/2 obtained in the case of a fixed-fixed right column for α = 1/3 and 10/3 Case

Column 1

α = h1 /h2

T.(EI/mh3 )1/2 hand calculus

T.(EI/mh3 )1/2 simplified calculus

T.(EI/mh3 )1/2 determinant

5

P-F

0.3333

0.6515

0.6980

0.6515

6

P-F

3.3333

1.8077

1.8138

1.8077

7

F-F

0.3333

0.3427

0.3490

0.3428

8

F-F

3.3333

1.7898

1.8138

1.7898

2.2 Right Column Fixed-Fixed The calculus of the vibration periods T.(EI/mh32 )1/2 for α = 1/3 and 10/3 in the case of a fixed-fixed right column is carried out similarly to Sect. 2.1. The results are summarized in Table 2. As expected, the analysis of the 1-DOF for high values of h1 leads to a behavior principally governed by the second column stiffness. T.(EI/mh3 )1/2 takes the value of 1.8138 when the first column effect is not considered.

3 Effect of a Torsion Spring on the Vibration Period The vibration analysis of an SDOF representing a rigid beam supported by two unequal columns is investigated. In order to take account of the torsion spring stiffness attached to the first column base a parameter (K *r ) is introduced, as follows: K r∗ = K r

h1 E.I

(5)

The study consists of varying both the heights ratio and the torsion spring stiffness for different ends conditions of the second column.

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Τ.(EI/mh )

3 1/2

2,5

*

Kr =0 * r * r * r * r * r * r * r * r * r

K =0.05

2,0

K =0.2 K =0.5

Kr

1,5

K =1.0 K =2.0 K =4.0

1,0

K =10.0 K =80.0

0,5

K =inf

0,0 0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

h1/h2 Fig. 4 Variations of T.(EI/mh3 )1/2 as a function of α and the parameter K *r in the case of a P-F second column

3.1 Right Column Pinned-Fixed After carrying out the eigenvalues problem-solving by using the null determinant search for different α and K *r values, the vibration period is extracted. The results are summarized under the form of the plotted curves of Fig. 4. These latter depend on both the heights ratio α and the dimensionless parameter K *r with a pinned-fixed second column. The curves enable a quick practical calculation of T.(EI/mh3 )1/2 . As shown, for α = 1/3 the value of the period is comprised between 0.3475(m.h32 /EI)1/2 and 0.6856(m.h32 /EI)1/2 . With the increase of the heights ratio α the T.(EI/mh3 )1/2 value increases until reaching a value of about 3.5 with α = 10/3. At this stage, the dynamic behavior of the system is only assured by the right column stiffness. It indicates a nil effect of the torsion spring.

3.2 Right Column Fixed-Fixed Similarly to the presentation of Sect. 3.1, a series of calculus are made for a fixedfixed right column. Figure 5 shows the T.(EI/mh3 )1/2 variations in function of α for variable values of the parameter K *r .

Vibration of an SDOF Representing a Rigid Beam …

159

2,0 1,8 1,6 *

K r =0

1,4

*

K r =0.05 *

K r =0.2

Τ.(EI/mh )

3 1/2

1,2

*

K r =0.5

Kr

*

K r =1.0

1,0

*

K r =2.0 *

K r =4.0

0,8

*

K r =10.0 *

0,6

K r =80.0 *

K r =inf

0,4 0,2 0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

h1/h2 Fig. 5 T.(EI/mh3 )1/2 variations in function of the ratio α and the K *r parameter in the case of a F-F second column

For all the plotted curves, an increase of values of the torsion spring stiffness drags the curves downwardly. For high heights ratios, the T.(EI/mh3 )1/2 is not affected by an increase of the spring stiffness. Thus, the dimensionless parameter T.(EI/mh3 )1/2 characterizing the vibration period takes a value of about 1.8, independently of the K *r values. This means that the lateral resistance is ensured by the right column only.

4 Conclusion The vibration analysis of a 1-DOF representing a rigid beam supported by two unequal columns with one having its lower end spring-hinged is investigated. The final results are shown on plotted curves representing the variations of the vibration period in function of both the heights ratios and the torsion spring stiffness. The presence of the torsion spring may be easily tackled only in the cases of little and high values of the heights ratios where a simplified hand calculus allows accurate results. Otherwise, the plotted curves are helpful to properly assess the sought vibration period T.

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As a further research, it will be more useful to include the shear effect to better reflect a short column behavior. This will concern the first column for little values of the h1 /h2 and the second column acting as a short column for higher values of that ratio.

References 1. Bathe, K.-J.: Finite Element Procedures. Prentice-Hall, Englewood Cliffs, NJ (2014) 2. Bathe, K.-J., Wilson, E.L.: Solution methods for eigenvalue problems in structural mechanics. Int. J. Numer. Meth. Eng. 6, 213–226 (1973). https://doi.org/10.1002/nme.1620060207 3. Clough, R.W., Penzien, J.: Dynamics of Structures. McGraw-Hill, New York (1975) 4. Liu, W.H., Wu, J.-R., Huang, C.-C.: Free vibration of beams with elastically restrained edges and intermediate concentrated masses. J. Sound Vib. 122, 193–207 (1988). https://doi.org/10. 1016/S0022-460X(88)80348-1 5. Maurizi, M.J., Rossi, R.E., Reyes, J.A.: Vibration frequencies for a uniform beam with one end spring-hinged and subjected to a translational restraint at the other end. J. Sound Vib. 48, 565–568 (1976). https://doi.org/10.1016/0022-460X(76)90559-9 6. Paz, M.: Structural Dynamics, Theory and Computation. Van Nostrand Reinhold Environmental Engineering Series. Van Nostrand Reinhold, New York (1980) 7. Rao, C.K., Mirza, S.: A note on vibrations of generally restrained beams. J. Sound Vib. 130, 453–465 (1989). https://doi.org/10.1016/0022-460X(89)90069-2

Vibration Analysis of a Uniform Beam Fixed at One End and Restrained Against Translation and Rotation at the Second One Farid Chalah, Lila Chalah-Rezgui, Salah Eddine Djellab and Abderrahim Bali

Abstract The problem of analyzing the transverse vibrations of uniform beams was largely reported in the scientific literature. The purpose of this work is to study the particular case of a uniform beam fixed at its left end and elastically restrained at the right end by two springs K T and K r . The latter are relative to the vertical translation and the rotation. The investigation consists of analyzing the influence of K T and K r on the uniform beam vibration behavior. The situations of extreme values of the two springs representing free, pinned, guided and fixed conditions at the right end served for the validation for which a good agreement was noted. Thereafter, it was extended to take into account the effect of the two springs acting simultaneously. The results are summarized under the form of plotted curves representing the variations of the angular vibration frequency ω1 as a function of both the K T and K r values. Keywords Angular vibration frequency · Beam vibration · Torsion spring · Translational spring · FEM

1 Introduction The problem of extracting the fundamental angular frequency ω1 of a vibrating uniform beam was often dealt with in the scientific literature. For this purpose, both theoretical and numerical methods were used. Concerning the transverse vibrations of beams, different methods are commonly applied [1, 5, 7]. This is also the case where the beams carry multiple point masses and spring-mass systems [4, 6, 8]. The existence of elastic springs constitutes similar situations for which the FEM was applied [3]. When investigating the transverse vibration of a uniform beam, various situations are considered, such as the existence of discrete masses attached or not to springs differently distributed along the studied beam. Many studies have been performed for analyzing the effect of placing translational or hinged springs F. Chalah (B) · L. Chalah-Rezgui · S. E. Djellab Faculty of Civil Engineering, Usthb, 16111 Algiers, Algeria A. Bali Ecole Nationale Polytechnique, Algiers, Algeria © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_15

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at particular locations. The method used to tackle this problem in the present study is the finite element method (FEM). For this purpose, a program written in the Fortran language is used to analyze the transverse vibration of the studied beam. The fundamental vibration circular frequency ω1 is extracted by using the Jacobi method. Although all the eigenvalues could be found for this problem, only the fundamental one was concerned in the present investigation. The FEM is a widely used method in structural and dynamic analyses [2]. It consists of discretizing the studied structure in elementary finite elements. The elementary stiffness and mass matrices are then assembled to form the global system matrices. Depending on the addressed problem, the displacements, stresses or eigenvalues are determined. This work focuses on providing practical curves to assess ω1 of an oscillating uniform beam (with its uniformly distributed mass) fixed at its left end and elastically restrained against translation and rotation at its right end. These plotted curves allow identifying the immediate effects of the elastic restraints on the dynamic behavior of the system. The present investigation allows an intuitive comprehension of beams vibration analysis when introducing flexible restraints.

2 Beam Modeling The free vibration analysis of a cantilever beam with torsional and translational springs located at its right end is investigated. The translational spring constitutes an elastic support at the right beam end and the rotational spring links that end to a mass less sliding system. The rotational spring is placed between the right beam end and the guided part to ensure a rotational restraint of the same right beam end. The final obtained system results in a uniform beam fixed at its first end and restrained in translation and rotation at the second, as shown in Fig. 1. The total beam of length L is divided into N elementary beams and the global stiffness and mass matrices of the system are obtained by assembling the N elementary ones. In this analysis, the first angular frequency ω1 is the only required one. Kr

.A E.I

L

KT

Fig. 1 Representation of a uniform beam fixed at one end and restrained in translation and rotation at the other (E.I—Product of the material Young’s modulus by the inertia; L—Beam length; ρ— Density of the material; A—Beam cross-section; K r —Torsion spring used to restraint the beam in rotation at its right node; K T —Vertical translation spring located at the right beam end)

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2.1 Elementary Stiffness Matrix The elementary stiffness matrix of a beam element of L length, based on the assumptions of the Euler-Bernoulli beam theory, is: ⎡ ⎢ [K ] = E.I.⎢ ⎣

12 6 −12 6 L3 L2 L3 L2 6 4 −6 2 L2 L L2 L −12 −6 12 −6 3 2 3 L L L L2 6 2 −6 4 2 2 L L L L

⎤ ⎥ ⎥ ⎦

(1)

2.2 Elementary Mass Matrix The consistent form of the mass matrix has been considered for the elementary beam element as follows: ⎡

156 ρ AL ⎢ 22L ⎢ [M] = 420 ⎣ 54 −13L

⎤ 22L 54 −13L 4L 2 13L −3L 2 ⎥ ⎥ 13L 156 −22L ⎦ −3L 2 −22L 4L 2

(2)

3 Analysis of the Beam Vibration for Extreme Values of K T and K r In the matrix formulation of the problem, the values of the stiffness (translation and torsion springs) values occupy the diagonal position in the global stiffness matrix corresponding to the translation and rotation terms, respectively, of the right node of the last elementary beam. The dimensionless parameters K T * = K T .L 3 /E.I and K r * = K r .L/E.I are introduced in what follows. The used method to look for the eigenvalues of Eq. 3 is that of Jacobi. [K ]{φ} = ω2 [M]{φ} where: ω angular vibration frequency. {φ} eigenvector.

(3)

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The preliminary validations concern the cases of extreme values of K T * and K r * . Different states of restraint conditions are obtained for each variation of K T * and K r * values. In these conditions, the possible conditions of the right node are: v = 0 and v = 0 for a free right node (K r * = 0, K T * = 0), see Fig. 2a. v = 0 and v = 0 for a hinged or pinned right node (K r * = 0, K T * = ∞), see Fig. 2b. v = 0 and v = 0 for a sliding or guided right node (K r * = ∞, K T * = 0), see Fig. 2c. v = 0 and v = 0 for a fixed or clamped right node (K r * = ∞, K T * = ∞), see Fig. 2d. where the first, second and third derivatives of the transverse displacement v are the slope, bending moment and shear force, respectively. Table 1 shows the values of the dimensionless angular frequency parameter α 2 = ω1 .(ρ.A.L 4 /E.I)1/2 for a comparison aim between the results obtained by the present study and theoretical results.with:  ω1 = α 2

E.I m b .L 4

(4)

where: α 2 Dimensionless factor characterizing the beam restraint conditions. mb linear distribution of the beam mass.

(a)Fixed-Free beam (Kr*=0, KT*=0)

(b)Fixed-Pinned beam(Kr*=0, KT*= ∞)

(c)Fixed-Guided beam(Kr*= ∞,KT*=0)

(d)Fixed-Fixed beam(Kr*=∞, KT*=∞)

Fig. 2 Final resulting ends conditions depending on the K r * and K T * extreme values Table 1 Values of α 2 for extreme K T * and K r * values K r * = 0.001

K r * = 0.001

K r * = 1.0d16

K r * = 1.0d16

K T * = 0.001

K T * = 1.0d16

K T * = 0.001

K T * = 1.0d16

Present study

3.5177

15.4193

5.5935

22.3733

Theory

3.5156

15.418

5.5933

22.3733

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3.1 Beam Vibration Analysis in the Case of KT * = 0 The case where K T * = 0 corresponds to Fig. 3. The curve plotted in Fig. 4 shows the variation of the dimensionless factor of the angular frequency ω1 .(ρ.A.L 4 /E.I)1/2 versus rotational spring K r * effect. In this situation, the translational spring is not considered (K T * = 0). In the interval (0.02 < K r * < 100), the torsional spring influences clearly the beam vibration behavior. For small of K r * , the behavior of the beam remains as a cantilever beam. This situation concerns also high values of K r * for which the beam becomes as a fixed-sliding beam.

Kr

Fig. 3 Representation of a uniform beam fixed at one end with a torsion spring at the other (K r * = 0, K T * = 0) 6,0

5,0

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1E-4

1E-3

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Fig. 4 Variation of the function of K r * for K T * = 0

dimensionless

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3.2 Beam Vibration Analysis in the Case of K T ∗ = ∞ In Fig. 5, the considered parameter K T * characterizing the translational spring value is high. It corresponds to a restraint of the vertical translation of the right end. Thus, the vertical elastic support becomes rigid. This makes the beam ends conditions fixed-pinned with a hinged-spring. Figure 6 shows the variations of the dimensionless angular frequency ω1 .(ρ.A.L 4 /E.I)1/2 parameter as a function of the rotational spring represented by K r*. Similar remarks to Sect. 3.1 can be formulated. For small values of K r * the studied beam behavior remains as that of a fixed-pinned beam. For intermediate values of

Fig. 5 Representation of a uniform beam fixed at its left end and restrained in rotation at the second end with K T * = ∞ 23 22 21

Kr

19 18

1

4

.( .A.L /E.I)

1/2

20

17 16 15 1E-4

1E-3

0,01

0,1

1

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Kr* Fig. 6 Variation of the dimensionless angular frequency parameter ω1 .(ρ.A.L 4 /E.I)1/2 as a function of K r * (for K T * = ∞) with K r * = K r .L/E.I

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K r * comprised between (0.02 and 800), K r * influences greatly the beam behavior which is considered fixed-partially hinged. For high values of K r * the studied beam becomes as fixed-fixed.

3.3 Beam Vibration Analysis in the Case Where K r ∗ = 0 or K r ∗ = ∞ The two curves of Fig. 7 are plotted by affecting the extreme values 0 and ∞ to K r * (characterizing the torsion spring) and varying the dimensionless parameter K T * representing the translational spring (with K T * = K T .L 3 /E.I). By increasing the rigidity of the translational spring, a progressive increase in the angular frequency is obtained in each curve of Fig. 7. For K r * = 0, the right end vertical restraint condition changes the beam behavior from free node K T * = 0, to a pinned node with K T * = ∞. In the upper curve of Fig. 7 corresponding to (K r * = ∞), the beam restraints vary from fixed-guided ends conditions to fixed-fixed at the right side. 24 22 *

E.I

18

L

Kr =0

Kr

.A

20

*

Kr =inf

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.( .A.L /E.I)

1/2

16

12

KT

1

10 8

KT

6 4 2 0 1E-3

0,01

0,1

1

10

KT

100

1000

10000

*

Fig. 7 Variations of the dimensionless angular frequency parameter as a function of K T * with K T * = K T .L 3 /E.I for K r * = 0 and K r * = ∞

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4 Effect of K T * and K r * on the Beam Angular Vibration Frequency Thereafter, the study is conducted in order to demonstrate the influence of both the two springs on the transverse vibrations of the considered uniform beam such restrained. Figure 8 represents the variation of ω1 .(ρ.A.L 4 /E.I)1/2 as a function of K T * (K T * = K T .L 3 /E.I) and K r * (K r * = K r .L/E.I) as a parameter. Five zones appear principally in Fig. 8, they correspond to K T * < 1, 1 < K T * < 10, 10 < K T * < 100, 100 < K T * < 1000 and beyond 1000. For K T * < 1, the dimensionless angular frequency parameter variation is to be linked principally to the K r * values whose limits vary between 3.5156 and 5.5933 while K T * implies a progressive increase. All the drawn curves for 1 < K T * < 10 take a curvilinear form and end up merging. Their variation becomes depending of K T * only for (10 < K T * < 100). Around the value of K T * = 50, the dimensionless angular frequency parameter value is proportional (in a logarithm representation) to K T * . Figure 8 shows a growth in the interval 100 < K T * < 1000 of all the curves. For high values of K T * (beyond K T * = 1000), the main influencing parameter is K r * . Then the values of ω1 .(ρ.A.L 4 /E.I)1/2 are comprised in the interval 15.418–22.373. 24 22

.A

20

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16 14

*

Kr =0.001 *

Kr =1

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*

Kr =3 *

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.( .A.L /E.I)

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*

Kr =10 *

Kr =20

8

*

Kr =40 *

Kr =80

6

*

Kr =200 *

Kr =inf

4 2 1E-3

0,01

0,1

1

10

KT

100

1000

10000

*

Fig. 8 Variation of ω1 .(ρ.A.L 4 /E.I)1/2 as a function of K T * (K T * = K T .L 3 /E.I) and K r * (K r * = K r .L/E.I) as a parameter

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5 Conclusion The problem of analyzing the vibration of a cantilever beam was largely examined and reported in the literature. In the present study, it is tackled by considering translation and rotational springs at its right end. As shown, the classical beams, namely fixedguided, fixed-free, fixed-pinned and fixed-fixed are obtained by considering for K r * and K T * the values of 0 and infinite. By changing progressively their values, the uniform beam behavior undergoes intermediate states describing various ends conditions at the right node. The proposed curves allow the calculation of the dimensionless angular frequency parameter ω1 .(ρ.A.L 4 /E.I)1/2 for the different values of K r * and K T * .

References 1. Banerjee, J.R.: Free vibration of beams carrying spring-mass systems—a dynamic stiffness approach. Comput. Struct. 104–105, 21–26 (2012). https://doi.org/10.1016/j.compstruc.2012. 02.020 2. Bathe, K.-J.: Finite Element Procedures. Prentice-Hall, Englewood Cliffs, NJ (2014) 3. Chalah-Rezgui, L., Chalah, F., Djellab, S.E., Nechnech, A., Bali, A.: Free vibration of a beam having a rotational restraint at one pinned end and a support of variable abscissa. In: Öchsner, A., Altenbach, H. (eds.) Mechanical and Materials Engineering of Modern Structure and Component Design, pp. 393–400. Springer International Publishing, Cham (2015). https://doi.org/10.1007/ 978-3-319-19443-1_32 4. Darabi, M.A., Kazemirad, S., Ghayesh, M.H.: Free vibrations of beam-mass-spring systems: analytical analysis with numerical confirmation. Acta. Mech. Sin. 28, 468–481 (2012). https:// doi.org/10.1007/s10409-012-0010-1 5. Lai, H.-Y., Hsu, J.-C., Chen, C.-K.: An innovative eigenvalue problem solver for free vibration of Euler-Bernoulli beam by using the Adomian decomposition method. Comput. Math. Appl. 56, 3204–3220 (2008). https://doi.org/10.1016/j.camwa.2008.07.029 6. Lin, H.-Y., Tsai, Y.-C.: Free vibration analysis of a uniform multi-span beam carrying multiple spring-mass systems. J. Sound Vib. 302, 442–456 (2007). https://doi.org/10.1016/j.jsv.2006. 06.080 7. Liu, Y., Gurram, C.S.: The use of He’s variational iteration method for obtaining the free vibration of an Euler-Bernoulli beam. Math. Comput. Model. 50, 1545–1552 (2009). https://doi.org/10. 1016/j.mcm.2009.09.005 8. Wu, J.-S., Hsu, T.-F.: Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered. Int. J. Mech. Sci. 49, 834–852 (2007). https://doi.org/10.1016/j.ijmecsci.2006.11.015

Fundamental Vibration Periods of Continuous Beams with Two Unequal Spans Farid Chalah, Lila Chalah-Rezgui, Salah Eddine Djellab and Abderrahim Bali

Abstract The analysis of the vibration of continuous beams was largely treated in the scientific literature by researchers using different methods. The objective of this study is to determinate the fundamental transverse vibration period T 1 of twospan beams having an intermediate support of varying abscissa. Various boundary conditions at the extreme ends of the analyzed continuous beams are considered. The analysis was conducted by the finite element method based upon the Euler– Bernoulli assumptions. The transverse vibration period is calculated for each position of the intermediate support for different end restraints. The validations are relative to particular abscissas and extreme locations of the intermediate support. Accurate results are obtained in accordance with the theory. At the end, the obtained results for the analyzed beams are plotted as curves representing the variations of T 1 as a function of the intermediate support location expressed by the ratio of the first span length over the total beam length. Keywords Continuous beam · Two-span beam · Transverse vibration · Unequal spans

1 Introduction The vibration of continuous beams is regularly investigated. It is tackled in the present study in the case of two unequal spans, having vertical support occupying various locations, for evaluating the fundamental vibration period T 1 . The considered vibration is relative to the vertical movement. Many works dealt with this vibration problem, see [4, 6]. This concerns also the efficient placement of rigid supports using finite element models [3].

F. Chalah (B) · L. Chalah-Rezgui · S. E. Djellab Faculty of Civil Engineering, Usthb, Algiers, Algeria A. Bali Ecole Nationale Polytechnique, Algiers, Algeria © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_16

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In this context, different procedures are used either theoretical or numerical approaches [1, 8]. Adding to this, some problems were considered using other methods as reported in [7] by the transfer matrix method when analyzing the case of pinned-pinned extreme ends. For this purpose, a Fortran program based on the finite element method (FEM) was written to treat the posed problem wherein a subroutine using the Jacobi method was implemented for extracting the fundamental vibration period T 1 .

2 Beam Modeling For analyzing the vibration of the continuous beam of a total length L constituted by two unequal spans, a total number of N = 64 elementary beams is utilized. The resulting stiffness and mass matrices size are 128 × 128. These two matrices are constructed by assembling the contributions of all the elementary beams’ stiffness and mass matrices. The eigenvalues problem solving is carried out for each location of the vertical rigid support with a step of L/N = 1/64.L. The used stiffness matrix of the elementary beam given in Eq. 1 is based on the assumptions of the Euler–Bernoulli beam theory: ⎡ ⎢ ⎢ [K e ] = E.I.⎢ ⎢ ⎣

12 L 3e 6 L 2e −12 L 3e 6 L 2e

6 −12 6 L 2e L 3e L 2e 4 −6 2 L e L 2e L e −6 12 −6 L 2e L 3e L 2e 2 −6 4 L e L 2e L e

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(1)

where E Young’s modulus. I inertia. L e elementary beam length. The elementary mass matrix for a uniformly distributed mass in its consistent form is used as written in Eq. (2): ⎡

156 ρ AL e ⎢ 22L e ⎢ [Me ] = 420 ⎣ 54 −13L e where ρ.A mass per unit length.

22L e 54 4L 2e 13L e 13L e 156 −3L 2e −22L e

⎤ −13L e −3L 2e ⎥ ⎥ −22L e ⎦ 4L 2e

(2)

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The resulting eigenvalues’ problem obtained given in Eq. (3) after assembling the elementary matrices and imposing the ends restraining conditions is solved by implementing the Jacobi method. [K ]{φ} = ω2 [M]{φ}

(3)

where ω angular vibrations frequency. {φ} eigenvector corresponding to its ω. The extracted eigenvalues ω1 –ωN are determined. In the present investigation, the only regarded value is that of ω1 which gives the sought fundamental period T 1 = 2π /ω1 of the free oscillation of a continuous beam. For each treated situation, the result is expressed in terms of the dimensionless parameter β = T 1 .(E.I/ρ.A.L 4 )1/2 .

3 Analysis of a Two-Span Beam in the Case of a Fixed First End In this section, the first end boundary condition is fixed and four second end restraints are considered: fixed, pinned, guided (sliding) or free. The first span length is noted (a) as shown after in all the figures presenting the two-span beam case treated with its boundary conditions. The resulting values of β as a function of a/L are given under plotted curves allowing to see the influence of the rigid support location on the dynamic behavior of the idealized vibrating depending on the two-end restraining conditions.

3.1 Second End Fixed The first investigated beam case is constituted by two unequal spans in the case of two fixed ends, as given in Fig. 1. Figure 2 shows the variation of β as a function of the first span length over the total beam length (a/L).

a

L

Fig. 1 Representation of a beam with two unequal spans in the case of two fixed ends

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a

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0,22

β

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L/2

L/2

0,12 0,10 0,08 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

a/L Fig. 2 Variation of β = T 1 .(E.I/ρ.A.L 4 )1/2 as a function of the first span length over the tot beam length (a/L) in the case of two clamped ends

As expected, the curve is symmetric. The studied beam behavior varies from a clamped-clamped beam of single span (for the support located at a = 0 or a = L) to a continuous beam with unequal spans. The value of β for a clamped-clamped beam is 0.28 which corresponds to the exact value of 2.π/4.7302 = 0.2808 extracted from [5].

3.2 Second End Pinned The second investigated beam of Fig. 3 is constituted by two unequal spans in the case of a first fixed end, while the second is pinned. Figure 4 highlights the various values that β takes by varying the locations of the vertical support in the case where the first end is fixed and the second end pinned.

a L Fig. 3 Representation of a beam with two unequal spans in the case where the first end is fixed and the second pinned

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0,45 0,40 0,35 a

L

β

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L/2

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L/2

0,15 0,10 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

a/L Fig. 4 Variation of β as a function of a/L in the case where the first end is fixed and the second pinned

For a = 0, the value of β corresponds to a fixed-pinned beam of L length. When the support is located near the right end (a ≈ L), it behaves as an almost clampedclamped beam. It takes the form of a clamped-pinned beam for support located at the right end (a = L). In the case of a clamped-pinned beam, the exact value of β is 2.π/3.9272 = 0.4074 according to [5].

3.3 Second End Guided The third analyzed beam of Fig. 5 is constituted by two unequal spans in the case of a first fixed end, while the second is guided (sliding). Figure 6 gives the different values of β for different abscissas of the considered support for the situation of a first clamped end with a second guided.

a

L

Fig. 5 Representation of a beam with two unequal spans in the case where the first end is fixed and the second guided

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1,0

a

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0,8

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0,2 0,0

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0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

a/L Fig. 6 Variation of β as a function of a/L in the case where the first end is clamped and the second guided (sliding)

The existence of a guided end is not usually tackled, but it completes the investigation. The particular abscissa of the support a = 0 does not affect the behavior of a clamped-guided beam. For values of a close to L (a/L ≈ 1) or a support location at a = L, the vertical translation is restrained and then the behavior of the right end is changed to a fixed one.

3.4 Second End Free The fixed-free beam with a support located at an abscissa (a) is shown in Fig. 7. This study case was considered for analyzing the free vibration of a cantilever beam with an intermediate support that results in an overhang of arbitrary length, as reported in [2].

a

L

Fig. 7 Variation of β as a function of a/L in the case where the first end is clamped and the second free

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2,0 1,8 1,6 a

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L/2

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0,2

0,3

0,4

0,5

0,6

0,7

0,8

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a/L Fig. 8 Variation of β as a function of a/L in the case of a fixed-free beam

Figure 8 gives the variation of β as a function of a/L in the case of a fixed-free beam. When the support is located at the left, the beam behavior remains as a cantilever beam with an exact β value of 2.π/1.8752 = 1.7872 deduced from its corresponding exact angular frequency, see [5]. The last location of the support (a = L) makes the initially fixed-free beam becoming a fixed-pinned beam.

4 Analysis of a Two-Span Beam in the Case of a Pinned First End This section is relative to the case where the first end is pinned, while the second one is either pinned or guided.

4.1 Second End Pinned In this subsection, the analyzed continuous beam of Fig. 9 is constituted by two unequal spans in the case of two pinned ends.

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Fig. 9 Representation of a continuous two-span beam having a vertical support at an abscissa a

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0,8

0,9

1,0

a/L Fig. 10 Variation of β as a function of the first span length over the total beam length (a/L) in the case where the two ends are pinned

Figure 10 shows the variations of β for a continuous two-span beam in function of the first span length over the total beam length a/L in the case of the two pinned extreme ends. When the support occupies the extreme locations a = 0 or a = L, the two-span beam behave as a simply supported beam and then β = 2.π/3.14162 = 0.6366, given from ref [5]. For abscissas of the support close to the locations a ≈ 0 or a ≈ L, the concerned extremity becomes as a fixed end. The studied beam with a single span is to be considered then as fixed-pinned beam (a ≈ 0) or pinned-fixed beam (a ≈ L). The curve of Fig. 10 is similar to that relative to the first mode given in [4].

4.2 Second End Guided The effect of the vertical support on the vibration period is investigated in the case where the right end is guided, as shown in Fig. 11.

Fundamental Vibration Periods of Continuous Beams …

a

179

L

Fig. 11 Representation of a pinned-guided continuous beam having a vertical support at an abscissa a

2,5

2,0

a

L

β

1,5

1,0 L/2

L/2

0,5

0,0 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

a/L Fig. 12 Variation of β as a function of the first span length over the total beam length a/L in the case where the first end is pinned and the second guided (sliding)

Figure 12 highlights the various behaviors of the continuous beam depending on the locations of the intermediate support. As shown for a location of the support close to the left end, the behavior of the twospan beam is as that of a fixed-guided beam of single span. For a = 0, the continuous beam behaves as pinned-guided one span beam. When, the support abscissa is at the right extreme end (the vertical displacement is restrained making that support fixed), the beam is to be considered as pinned-clamped of one span.

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5 Two Ends Guided In this section, the case study concerns two guided ends, as shown in Fig. 13. Figure 14 illustrates the curve variations giving the values of β in function of the span length a relatively to the total continuous beam length L. The symmetry of the curve indicates that the two ends such restrained induce a similar behavior. By restraining vertically the sliding possibility of a given node, its nature is changed into a fixed end. This situation concerns both the two similar ends (guided).

a

L

Fig. 13 Representation of a continuous two-span beam with guided ends having a vertical support at an abscissa a 1,2 1,1

β

1,0

a

L

0,9 0,8

L/2

L/2

0,7 0,6 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

a/L Fig. 14 Variation of β as a function of the first span length over the total continuous beam length (a/L) in the case where the two ends are guided (sliding)

Fundamental Vibration Periods of Continuous Beams …

181

6 Conclusion For any intermediate location of the support, a reading operation gives the value of the fundamental vibration period in term of the dimensionless parameter β = T 1 .(E.I/ρ.A.L 4 )1/2 for the studied two-span beam in function of a/L depending on its extreme ends boundary conditions. It is to be noted that the minimal values of β in all the plotted curves correspond to the second vibration mode of single-span beams of L length (without intermediate support) with the same two-end boundary conditions.

References 1. Bathe, K.-J.: Finite Element Procedures, 2nd edn. Prentice-Hall, Englewood Cliffs, NJ (2014) 2. Chalah-Rezgui, L., Chalah, F., Falek, K., Bali, A., Nechnech, A.: Transverse vibration analysis of uniform beams under various ends restraints. APCBEE Procedia 9, 328–333 (2014). https:// doi.org/10.1016/j.apcbee.2014.01.058 3. Friswell, M.I.: Efficient placement of rigid supports using finite element models. Commun. Numer. Methods Eng. 22, 205–213 (2005). https://doi.org/10.1002/cnm.808 4. Gorman, D.J.: Free lateral vibration analysis of double-span uniform beams. Int. J. Mech. Sci. 16, 345–351 (1974). https://doi.org/10.1016/0020-7403(74)90008-3 5. Harris, C.M., Piersol, A.G. (eds.): Harris’ shock and vibration handbook. McGraw-Hill Handbooks, 5th edn. McGraw-Hill, New York (2002) 6. Laura, P.A.A., Sánchez Sarmiento, G., Bergmann, A.N.: Vibrations of double-span uniform beams subject to an axial force. Appl. Acoust. 16, 95–104 (1983). https://doi.org/10.1016/0003682X(83)90032-4 7. Liu, T., Feng, F., Chen, Y.Z., Wen, B.C.: Research on vibration and instability of the double-span beam base on transfer matrix. Appl. Mech. Mater. 16–19, 160–163 (2009). https://doi.org/10. 4028/www.scientific.net/AMM.16-19.160 8. Zienkiewicz, O.C., Taylor, R.L., Zienkiewicz, O.C.: Solid mechanics. The Finite Element Method. Butterworth-Heinemann, 5th edn, reprinted. Oxford (2003)

Transverse Displacements of Transversely Cracked Beams with a Linear Variation of Width Due to Axial Tensile Forces Matjaž Skrinar

Abstract This paper considers the analysis of transverse displacements of slender beams with a linear variation of width and a single-sided transverse crack, subjected to axial tensile forces. When analysing such structures, the application of detailed 2D or 3D finite element meshes is undoubtedly the best solutions possible. However, in inverse identification of potential cracks such comprehensive models are actually not the most suitable solutions. Therefore, the presented studies implement a simplified model where the crack is represented by means of an internal hinge endowed with a rotational spring. In the first part of the research presented, solutions from the simplified model’s governing differential equations are obtained. The purpose of this study is to demonstrate the model’s ability to adequately describe the considered phenomenon and to derive an appropriate rotational spring definition. The second part of the research discussed is devoted to modelling of the phenomenon by the simple one-dimensional beam finite element. Afterwards, the implementation and the quality of the results are being presented through a comparative case study that complements the derivations. Keywords Beam finite element · Stiffness matrix and load vector · Linear variation of width · Transverse cracks · Transverse displacements · Axial tensile forces

1 Introduction Even if beams are rather simple structural elements, they also represent a very important and frequently used structural component. Generally, a beam is a straight bar of an arbitrary cross section, but the majority of research is focused onto beams with constant cross sections which are also the most frequent ones. The derivations of the stiffness matrix and the corresponding load vector due to transverse loads can

M. Skrinar (B) Faculty of Civil Engineering, Transportation Engineering and Architecture, University of Maribor, Maribor, Slovenia e-mail: [email protected] © Springer Nature Switzerland AG 2020 A. Öchsner and H. Altenbach (eds.), Engineering Design Applications III, Advanced Structured Materials 124, https://doi.org/10.1007/978-3-030-39062-4_17

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be found in numerous references (e.g. Smith and Griffiths [5] for non-cracked prismatic beam, Skrinar [3] for the cracked multi-stepped beams and beams with linearly varying heights). When considering beams’ displacements we usually individually consider axial displacements due to axial loads from the transverse displacements due to the transverse loads. Furthermore, in the situations where the transverse displacements are influenced by axial loads, we primarily focus on compressive axial forces, mainly due to the buckling problems. However, for beams with a single-sided transverse crack, the transverse displacements occur also due to axial tensile forces. This paper thus studies this phenomenon for beams with a linear variation of the width by applying a simplified model of the crack. Differential equations of the mathematical model are solved initially, followed by an approximate yet effective beam finite element solution.

2 Problem Description and Preliminary Studies In the preliminary phase, beams with a single-sided transverse crack differing in boundary conditions and loaded with axial tensile forces were studied. The length of the beams was 5 m, the height was h = 0.20 m and the width b was decreasing from bo = 0.5 m at the left-end to bL = 0.3 m at the right-end. Young’s modulus of the material was E = 30 GPa, and the axial load was 1 MN. Three cracks, located on the upper beam’s surface, were separately introduced at the equidistant locations (1.25, 2.5 and 3.75 m) from the left-end. All the beams were modelled by 3D finite elements computational models where the displacements and reactions were obtained from a computational model consisting of 48,800 3D solid elements with 74,538 nodes. Vertical and horizontal displacements of discrete nodal points were obtained by solving approximately 223,000 linear equations. The crack, which was precisely modelled by the discrete approach, tended to open simultaneously causing transverse displacements along with the element where their magnitudes increased with the crack depth. Figure 1a, b shows the distribution of transverse displacements in the form of discrete points along the beam for a simply supported beam and a propped cantilever, respectively. It should be noted that although the transverse displacements were obtained exclusively by tensile axial forces the mathematical functions are clearly different for both beam’s types. For the simply supported beam as well as the cantilever (not shown), the displacements evidently followed linear distribution while for the propped cantilever this distribution was parabolic.

Transverse Displacements of Transversely Cracked Beams …

(a) 0.005

185

v [m] δ=0.5

0.004 0.003

δ=0.4

0.002 δ=0.3 0.001

δ=0.2

0

(b)

1

2

δ=0.1

v [m]

x [m] 3

4

5

δ=0.5

0.0025 0.002

δ=0.4

0.0015 0.001

δ=0.3 0.0005

0

δ=0.2

1

2

δ=0.1

x [m] 3

4

5

Fig. 1 a Comparison of transverse displacements of the simply supported beam for various relative crack depths (δ = 0.1,…, 0.5), b comparison of transverse displacements of the propped cantilever for various relative crack depths (δ = 0.1,…,0.5)

3 Mathematical Model Formulation and Derivation of New Rotational Spring Stiffness 3.1 Formal Approach The general governing differential equation (GDE) of the elastic line for a slender beam subjected to bending in the plane of symmetry that relates transverse displacement v(x), the coordinate x, the geometrical and mechanical properties of the cross section (flexural rigidity EI(x)), axial force N x (x) and the applied transverse load q(x), known also as Euler–Bernoulli equation is given as:   d2 v(x) d2 v(x) d2 E I · − N · = q(x) (x) x dx 2 dx 2 dx 2

0≤x≤L

(3.1)

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When studying the beams with a linear variation of width the flexural rigidity also follows a linear distribution. The variation of flexural rigidity EI(x) within the beam is thus described as: E I (x) = E I o + k E I · x

(3.2)

with kE I =

EIL − EIo L

(3.3)

where EI o and EI L represent flexural rigidities at the start and at the end of the beam, respectively. Although Eq. (3.3) can be introduced into Eq. (3.2) prior to any further derivations, this leads to logarithmic terms with negative arguments in the derived solutions for simple bending causing numerical issues (Skrinar [4]). However, these problems are completely avoided by introducing the coefficient k EI which further yields more abbreviate forms of the derived at expressions. Therefore, Eq. (3.1) obtains the form: d4 v(x) d4 v(x) d3 v(x) + k · x · + 2 · k · E I E I dx 4 dx 4 dx 3 d2 v(x) − Nx · = q(x) 0 ≤ x ≤ L dx 2

E Io ·

(3.4)

However, the analytical solution of Eq. (3.4) could not be obtained (and furthermore, its solution for q(x) = 0 is v(x) ≡ 0).

3.2 Introduction of “Internal” Bending Moment MN Therefore, the decision was met to include the effect of tensile axial force N x to the crack in a different way. As the crack of depth d shifts the position of normal stresses’ resultant deeper into non-cracked part of the cross section, the pair of two equal axial forces (which act in opposite directions but not through the same point) produces an effect of a couple which opens the crack, as shown in Fig. 2. The actual moment (or lever) arm depends on the correct distribution of normal stresses in the longitudinal direction which is rather complex. Therefore, in order to simplify Fig. 2 Presentation of internal sagging bending moment M N due to the pair of two equal forces

MN

Nx

Nx

d 2

Nx d

MN

Nx

h

Transverse Displacements of Transversely Cracked Beams …

187

the computation, the moment arm was taken as the half of the crack depth d, and, consequently, allowing the “internal” sagging moment M N to be written simply as M N = N x · d/2. It should be noted that for the situation when the beam is cracked at the top surface a hogging bending occurs and, consequently, the negative sign must be implemented in the bending moment’s evaluation. By including the effect of axial force through the “internal” bending moment M N , Eq. (3.4) thus reduces into: EIo ·

d4 v(x) d4 v(x) d3 v(x) + k · x · + 2 · k · = q(x) E I E I dx 4 dx 4 dx 3

0 ≤ x ≤ L (3.5)

The general solution of this differential equation is given in the following mathematical form:   C1 E I (x) + C4 · x 2 − C1 · 3 · Ln(E I (x)) v(x) = C2 + x · C3 + 2 kE I kE I + particular integral

(3.6)

where C 1 , C 2 , C 3 and C 4 are constants of the integration obtained from boundary conditions, while the particular integral depends on the mathematical form of load q(x). It is instructive to note that this solution is not given in a form of simple polynomials as it is the case for the constant flexural rigidity EI. For the case under consideration, where the transverse displacements are considered as an exclusive effect of axial forces (i.e. q(x) = 0), the particular integral equals zero. However, the crack separates the beam into two elastic parts, and the transverse displacements cannot be described by a single function anymore. Therefore, two displacement functions for the parts on the left (v1 (x)) and right (v2 (x)) side of the crack are required. These functions are obtained from two coupled differential equations that have to be solved simultaneously. Their solutions—functions v1 (x) and v2 (x) for the parts to the left and the right, respectively, contain eight unknown constants altogether:   C1 + C4 · x 2 − C1 · v1 (x) = C2 + x · C3 + 2 kE I C1 − 2 · x · Ln(E I o + k E I · x) kE I   D1 v2 (x) = D2 + x · D3 + 2 + D4 · x 2 − D1 · kE I D1 − 2 · x · Ln(E I o + k E I · x) kE I

EIo · Ln(E I o + k E I · x) k 3E I (3.7) EIo · Ln(E I o + k E I · x) k 3E I (3.8)

The effect of the transverse crack located at the distance L 1 from the left-end is introduced through the simplified computational model as given by Okamura

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et al. [1]. In this model, each crack is represented by means of an internal hinge endowed with a rotational spring that takes into account the cross section’s residual stiffness. The rotational spring’ stiffness is denoted as K r and for its evaluation, several definitions can be found in the references. Four unknown constants are determined from mechanical and kinematical continuity conditions at the crack location (x = L 1 ), where the presence of the crack causes a slope discontinuity. These conditions are the equality of displacement: v1 (L 1 ) = v2 (L 1 ),

(3.9)

the condition for the discrete increase of rotations (which now also includes the effect of internal bending moment M N to the slope discontinuity): EI ·

d2 v1 (L 1 ) + M N = (ϕ2 (L 1 ) − ϕ1 (L 1 )) · K r N dx 2

(3.10)

the equality of bending moments: d2 v1 (L 1 ) d2 v2 (L 1 ) = dx 2 dx 2

(3.11)

and the equality of shear forces which reduce into: d3 v1 (L 1 ) d3 v2 (L 1 ) = 3 dx dx 3

(3.12)

Although several definitions for rotational spring stiffness exist, their implementation was questionable due to the simplification regarding the inner bending moment M N arm definition. Therefore, a new definition of the stiffness K rN was derived. Initially, transverse displacements’ functions were firstly obtained in analytical forms for three beams with different boundary conditions (cantilever, simply supported and propped cantilever). Afterwards, all known geometrical and mechanical data (except the values of K rN ) was inserted into beam’s specific equations of transverse displacements. Finally, the discrete rotational spring stiffness’ values of K rN were identified from the condition that the maximal displacement of the simplified 1D model should be equal to the corresponding value from the detailed 3D FE model. For each relative crack depth δ (from 0.1 to 0.5 in steps of 0.1), three locations on each beam (onequarter of the length, mid-span and three-quarters of the length) were considered, thus yielding altogether nine discrete values of K rN for each consider relative depth. The discrete identified values for δ = 0.5 are presented in Table 1. It is evident from the table that quite similar, but not perfectly identical values were identified for the presented relative depth (the difference between the maximal and minimal value is 1.31%). Therefore, their average value of 3.326316 was used for further analysis. This value underestimates the higher identified value by 0.66% and overestimates the smaller value by 0.64% (where the average discrepancy is smaller 0.002%).

Transverse Displacements of Transversely Cracked Beams …

189

Table 1 Identified discrete values of K rN for three different beams for δ = 0.5 Location (m)

Cantilever

Simply supported beam

Clamped—hinged

1.25

3.304416

3.311568

3.347675

2.5

3.314656

3.320090

3.333958

3.75

3.328567

3.335980

3.339938

After repeating the averaging process for the other relative depths five discrete values of K rN of an appropriate function were obtained. Its form was assumed in the same form as for similar functions due to standard bending and, consequently, just a matching function f N (δ) in denominator had to be evaluated, Eq. (3.13): Kr N =

EI h · f N (δ)

(3.13)

with f N (δ) = 2.757030 · δ + 3.660819 · δ 2 + 1.006341 · δ 3 − 4.262105 · δ 4 + 37.541938 · δ 5 The derived at expression was compared against functions from known similar definitions given by some other authors, Fig. 3, which shows similar behaviour of all existing definitions. 5 4

f(δ) newly derived function existing functions

3 2

fN(δ)

1 0

0.1

0.2

0.3

0.4

0.5

δ

1Fig. 3 Comparison of various denominators’ functions from existing rotational spring stiffness definitions

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4 Verifying the Obtained Definition Through Testing of Differential Equations Solutions To examine the proposed model’s behaviour, the GDEs were resolved for the considered “prime” beam utilising the newly derived rotational spring stiffness’ function K rN (δ) instead of the example specific ideal (i.e. identified) values. Their solutions were further compared to the 3D FE model solutions. The second and third columns of Tables 2, 3 and 4 thus present the examples’ maximal displacements from 3D FE models and GDEs, respectively. The matching of the results was good not only at the crack location but also overall along the complete simply supported beam, Fig. 4, where the transverse displacements’ functions were simple linear polynomials. Table 2 Comparison of transverse displacements at different crack’s locations for the simply supported beam

Table 3 Comparison of transverse displacements at the free end for different crack’s locations on the cantilever

Table 4 Comparison of transverse displacements at the free end for different crack’s locations for the propped cantilever

Crack location (m)

3D FE model (mm)

GDEs’ solutions (error)

1.25

3.44955

3.46491 mm (0.445%)

2.5

5.18764

5.19736 mm (0.188%)

3.75

4.46783

4.45489 mm (−0.290%)

Crack location (m)

3D FE model (mm)

GDEs’ solutions (error)

1.25

−13.7684

−13.8596 mm (0.663%)

2.5

−10.3583

−10.3947 mm (0.352%)

3.75

−5.94387

−5.9396 mm (−0.068%)

Crack location (m)

3D FE model (mm)

GDEs’ solutions (error)

1.25

0.90342

0.89871 mm (−0.522%)

2.5

2.78813

2.78238 mm (−0.206%)

3.75

3.58022

3.56607 mm (−0.395%)

Transverse Displacements of Transversely Cracked Beams …

191

v [m] 0.004 0.003 0.002

GDEs 48,800 3D FE

0.001

x [m]

0

1

2

3

4

5

Fig. 4 Comparison of transverse displacements of the simply supported beam from both applied models

The transverse displacements functions were simple linear polynomials also for the cantilever, where the matching of the results was again good for the complete element, as shown in Fig. 5. Also for the complete propped cantilever, the matching of the results was good along the complete beam, as shown in Fig. 6. However, it is clearly evident that the transverse displacements functions were not simple linear polynomials anymore.

0

1

2

3

4

5

x [m]

-0.002 -0.004 -0.006 -0.008 -0.01 -0.012

GDEs 48,800 3D FE

-0.014 v [m] Fig. 5 Comparison of transverse displacements of the cantilever from both applied models

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v [m] GDEs 48,800 3D FE

0.0025 0.002 0.0015 0.001 0.0005 0

x [m] 1

2

3

4

5

Fig. 6 Comparison of transverse displacements of the propped cantilever from both applied models

5 Beam Finite Element Solution 5.1 Derivation of Stiffness Matrix’ Coefficients The governing differential equation of bending of straight symmetric beams (I yz = 0), which can be found in many references (Reddy [2]) is given in the form:   d2 v(x) d2 E I · = q(x) (x) dx 2 dx 2

0