Shear stress solutions for curved beams: a structural analysis approach
The shear stress on isotropic curved beams with compact sections and variable thickness is investigated. Two new solutions, based on Cook’s proposal and the mechanics of materials approach, were developed and validated using computational finite element models (FEM) for four typical cross-sections (...
- Autores:
-
Palencia Díaz, Argemiro
Velilla Díaz, Wilmer
Contreras, Victor
Guillén-Rujano, Renny
Hernández-Pérez, Adrián
- Tipo de recurso:
- Fecha de publicación:
- 2024
- Institución:
- Universidad Tecnológica de Bolívar
- Repositorio:
- Repositorio Institucional UTB
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.utb.edu.co:20.500.12585/12952
- Palabra clave:
- Curved beams
Straight beams
Shear stress
Mechanics of materials
Theory of elasticity
LEMB
- Rights
- openAccess
- License
- http://creativecommons.org/publicdomain/zero/1.0/
| id |
UTB2_f89f2f7352ee9d7455e2c908852eac30 |
|---|---|
| oai_identifier_str |
oai:repositorio.utb.edu.co:20.500.12585/12952 |
| network_acronym_str |
UTB2 |
| network_name_str |
Repositorio Institucional UTB |
| repository_id_str |
|
| dc.title.spa.fl_str_mv |
Shear stress solutions for curved beams: a structural analysis approach |
| title |
Shear stress solutions for curved beams: a structural analysis approach |
| spellingShingle |
Shear stress solutions for curved beams: a structural analysis approach Curved beams Straight beams Shear stress Mechanics of materials Theory of elasticity LEMB |
| title_short |
Shear stress solutions for curved beams: a structural analysis approach |
| title_full |
Shear stress solutions for curved beams: a structural analysis approach |
| title_fullStr |
Shear stress solutions for curved beams: a structural analysis approach |
| title_full_unstemmed |
Shear stress solutions for curved beams: a structural analysis approach |
| title_sort |
Shear stress solutions for curved beams: a structural analysis approach |
| dc.creator.fl_str_mv |
Palencia Díaz, Argemiro Velilla Díaz, Wilmer Contreras, Victor Guillén-Rujano, Renny Hernández-Pérez, Adrián |
| dc.contributor.author.none.fl_str_mv |
Palencia Díaz, Argemiro Velilla Díaz, Wilmer Contreras, Victor Guillén-Rujano, Renny Hernández-Pérez, Adrián |
| dc.subject.keywords.spa.fl_str_mv |
Curved beams Straight beams Shear stress Mechanics of materials Theory of elasticity |
| topic |
Curved beams Straight beams Shear stress Mechanics of materials Theory of elasticity LEMB |
| dc.subject.armarc.none.fl_str_mv |
LEMB |
| description |
The shear stress on isotropic curved beams with compact sections and variable thickness is investigated. Two new solutions, based on Cook’s proposal and the mechanics of materials approach, were developed and validated using computational finite element models (FEM) for four typical cross-sections (rectangular, circular, elliptical, and triangular) used in civil and mechanical structures, constituting a novel approach to predicting shear stresses in curved beams. They predict better results than other reported equations, are simpler and easier for engineers to use quickly, and join the group of equations found using the theory of elasticity, thereby expanding the field of knowledge. The results reveal that both equations are suitable to predict the shear stress on a curved beam with outer/inner radii ratios in the interval 1 < b/a ≤5 aspect ratios. There is a maximum relative difference between the present solutions and finite element models of 8% within 1 < b/a ≤2, and a maximum of 16% in 2 < b/a ≤5. Additionally, the neutral axis of the curved beam can be located with the proposed solution and its position matches with that predicted by FEM. The displacement at the top face of the end of the curved beam induces a difference in the shear stress results of 8.0%, 7.0%, 6.5%, and 2.9%, for the circular, rectangular, elliptical, and triangular cross-sections, respectively, when a 3D FEM solution is considered. For small b/a ratios (near 1), the present solutions can be reduced to Collignon’s formula. |
| publishDate |
2024 |
| dc.date.accessioned.none.fl_str_mv |
2024-12-06T16:24:14Z |
| dc.date.available.none.fl_str_mv |
2024-12-06T16:24:14Z |
| dc.date.issued.none.fl_str_mv |
2024-12-06 |
| dc.date.submitted.none.fl_str_mv |
2024-12-06 |
| dc.type.coarversion.fl_str_mv |
http://purl.org/coar/version/c_970fb48d4fbd8a85 |
| dc.type.driver.spa.fl_str_mv |
info:eu-repo/semantics/article |
| dc.type.hasversion.spa.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
| dc.type.spa.spa.fl_str_mv |
http://purl.org/coar/resource_type/c_2df8fbb1 |
| status_str |
publishedVersion |
| dc.identifier.citation.spa.fl_str_mv |
Guillén-Rujano, R., Contreras, V., Palencia-Díaz, A., Velilla-Díaz, W., & Hernández-Pérez, A. (2024). Shear Stress Solutions for Curved Beams: A Structural Analysis Approach. Materials, 17(23), 5982. https://doi.org/10.3390/ma17235982 |
| dc.identifier.uri.none.fl_str_mv |
https://hdl.handle.net/20.500.12585/12952 |
| dc.identifier.doi.none.fl_str_mv |
https://doi.org/10.3390/ma17235982 |
| dc.identifier.instname.spa.fl_str_mv |
Universidad Tecnológica de Bolívar |
| dc.identifier.reponame.spa.fl_str_mv |
Repositorio Universidad Tecnológica de Bolívar |
| identifier_str_mv |
Guillén-Rujano, R., Contreras, V., Palencia-Díaz, A., Velilla-Díaz, W., & Hernández-Pérez, A. (2024). Shear Stress Solutions for Curved Beams: A Structural Analysis Approach. Materials, 17(23), 5982. https://doi.org/10.3390/ma17235982 Universidad Tecnológica de Bolívar Repositorio Universidad Tecnológica de Bolívar |
| url |
https://hdl.handle.net/20.500.12585/12952 https://doi.org/10.3390/ma17235982 |
| dc.language.iso.spa.fl_str_mv |
eng |
| language |
eng |
| dc.rights.coar.fl_str_mv |
http://purl.org/coar/access_right/c_abf2 |
| dc.rights.uri.*.fl_str_mv |
http://creativecommons.org/publicdomain/zero/1.0/ |
| dc.rights.accessrights.spa.fl_str_mv |
info:eu-repo/semantics/openAccess |
| dc.rights.cc.*.fl_str_mv |
CC0 1.0 Universal |
| rights_invalid_str_mv |
http://creativecommons.org/publicdomain/zero/1.0/ CC0 1.0 Universal http://purl.org/coar/access_right/c_abf2 |
| eu_rights_str_mv |
openAccess |
| dc.format.extent.none.fl_str_mv |
18 páginas |
| dc.format.mimetype.spa.fl_str_mv |
application/pdf |
| dc.publisher.place.spa.fl_str_mv |
Cartagena de Indias |
| dc.publisher.faculty.spa.fl_str_mv |
Ingeniería |
| dc.publisher.sede.spa.fl_str_mv |
Campus Tecnológico |
| dc.publisher.discipline.spa.fl_str_mv |
Ingeniería Mecánica |
| institution |
Universidad Tecnológica de Bolívar |
| bitstream.url.fl_str_mv |
https://repositorio.utb.edu.co/bitstream/20.500.12585/12952/1/materials-17-05982.pdf https://repositorio.utb.edu.co/bitstream/20.500.12585/12952/2/license_rdf https://repositorio.utb.edu.co/bitstream/20.500.12585/12952/3/license.txt https://repositorio.utb.edu.co/bitstream/20.500.12585/12952/4/materials-17-05982.pdf.txt https://repositorio.utb.edu.co/bitstream/20.500.12585/12952/5/materials-17-05982.pdf.jpg |
| bitstream.checksum.fl_str_mv |
c5c90530d810b22376701231ec5edbe2 42fd4ad1e89814f5e4a476b409eb708c e20ad307a1c5f3f25af9304a7a7c86b6 b8ad66064a9821d31ebfc02c62d1f62f f15bd18ce857387ddce567d8401cabf9 |
| bitstream.checksumAlgorithm.fl_str_mv |
MD5 MD5 MD5 MD5 MD5 |
| repository.name.fl_str_mv |
Repositorio Institucional UTB |
| repository.mail.fl_str_mv |
repositorioutb@utb.edu.co |
| _version_ |
1834107711106580480 |
| spelling |
Palencia Díaz, Argemirof16cdaf5-429b-4cc1-9893-7261733d2f39Velilla Díaz, Wilmer0b62bcbc-7077-4361-9708-424b68c21cf7Contreras, Victore22092e1-5f49-4b62-a596-6f84a0e65df0Guillén-Rujano, Rennye287fa99-eca4-44cf-a116-1c6fa5048a9cHernández-Pérez, Adrián3bea15f9-aba1-41e3-9ce9-d1a76082e2de2024-12-06T16:24:14Z2024-12-06T16:24:14Z2024-12-062024-12-06Guillén-Rujano, R., Contreras, V., Palencia-Díaz, A., Velilla-Díaz, W., & Hernández-Pérez, A. (2024). Shear Stress Solutions for Curved Beams: A Structural Analysis Approach. Materials, 17(23), 5982. https://doi.org/10.3390/ma17235982https://hdl.handle.net/20.500.12585/12952https://doi.org/10.3390/ma17235982Universidad Tecnológica de BolívarRepositorio Universidad Tecnológica de BolívarThe shear stress on isotropic curved beams with compact sections and variable thickness is investigated. Two new solutions, based on Cook’s proposal and the mechanics of materials approach, were developed and validated using computational finite element models (FEM) for four typical cross-sections (rectangular, circular, elliptical, and triangular) used in civil and mechanical structures, constituting a novel approach to predicting shear stresses in curved beams. They predict better results than other reported equations, are simpler and easier for engineers to use quickly, and join the group of equations found using the theory of elasticity, thereby expanding the field of knowledge. The results reveal that both equations are suitable to predict the shear stress on a curved beam with outer/inner radii ratios in the interval 1 < b/a ≤5 aspect ratios. There is a maximum relative difference between the present solutions and finite element models of 8% within 1 < b/a ≤2, and a maximum of 16% in 2 < b/a ≤5. Additionally, the neutral axis of the curved beam can be located with the proposed solution and its position matches with that predicted by FEM. The displacement at the top face of the end of the curved beam induces a difference in the shear stress results of 8.0%, 7.0%, 6.5%, and 2.9%, for the circular, rectangular, elliptical, and triangular cross-sections, respectively, when a 3D FEM solution is considered. For small b/a ratios (near 1), the present solutions can be reduced to Collignon’s formula.18 páginasapplication/pdfenghttp://creativecommons.org/publicdomain/zero/1.0/info:eu-repo/semantics/openAccessCC0 1.0 Universalhttp://purl.org/coar/access_right/c_abf2Shear stress solutions for curved beams: a structural analysis approachinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_2df8fbb1http://purl.org/coar/version/c_970fb48d4fbd8a85Curved beamsStraight beamsShear stressMechanics of materialsTheory of elasticityLEMBCartagena de IndiasIngenieríaCampus TecnológicoIngeniería MecánicaInvestigadoresWu, S.; Li, Y.; Bao, Y.; Zhu, J.; Wu, H. Examination of Beam Theories for Buckling and Free Vibration of Functionally Graded Porous Beams. Materials 2024, 17, 3080.Borković, A.; Marussig, B.; Radenković, G. Geometrically exact static isogeometric analysis of an arbitrarily curved spatial Bernoulli–Euler beam. Comput. Methods Appl. Mech. Eng. 2022, 390, 114447.Zhao, X.; Zhou, Y.; Shao, Y.B.; Liu, B.; Zhou, R. Analytical solutions for forced vibrations of Timoshenko curved beam by means of Green’s functions. Eng. Mech. 2020, 37, 12–27.Nicolalde, J.F.; Yaselga, J.; Martínez-Gómez, J. Selection of a sustainable structural beam material for rural housing in Latin América by multicriteria decision methods means. Appl. Sci. 2022, 12, 1393Zhang, S.; Qian, D.; Zhang, Z.; Ge, H. Low-Frequency Bandgap Characterization of a Locally Resonant Pentagonal Phononic Crystal Beam Structure. Materials 2024, 17, 1702.Zhang, P.; Qing, H.; Gao, C.F. Exact solutions for bending of Timoshenko curved nanobeams made of functionally graded materials based on stress-driven nonlocal integral model. Compos. Struct. 2020, 245, 112362.Bhagatji, J.D.; Kravchenko, O.G.; Asundi, S. Mechanics of Pure Bending and Eccentric Buckling in High-Strain Composite Structures. Materials 2024, 17, 796.Ham, S.; Ji, S.; Cheon, S.S. The design of a piecewise-integrated composite bumper beam with machine-learning algorithms. Materials 2024, 17, 602.Ye, S.Q.; Mao, X.Y.; Ding, H.; Ji, J.C.; Chen, L.Q. Nonlinear vibrations of a slightly curved beam with nonlinear boundary conditions. Int. J. Mech. Sci. 2020, 168, 105294.Song, D.; Kim, R.; Choi, K.; Shin, D.; Lee, S. Effects of Beam Shape on the Microstructures and Mechanical Properties during Thin-Foil Laser Welding. Metals 2023, 13, 916.He, X.T.; Wang, X.; Zhang, M.Q.; Sun, J.Y. The Thermal Stress Problem of Bimodular Curved Beams under the Action of End-Side Concentrated Shear Force. Materials 2023, 16, 5221.Cao, X.; Ni, J.; Shao, C.; Yang, X.; Lou, C. Nearly Perfect Transmission of Lamé Modes in a Rectangular Beam with Part and Through-Thickness Vertical Cracks. Materials 2023, 16, 4164Velilla-Díaz, W.; Pinzón, R.; Guillén-Rujano, R.; Pérez-Ruiz, J.; de Lacalle, L.; Palencia, A.; Maury, H.; Zambrano, H. Fatigue Life Estimation Model of Repaired Components with the Expanded Stop-Hole Technique. Metals 2024, 14, 182.Lindeburg, M.R.; Baradar, M. Seismic Design of Building Structures; Professional Publications, Inc.: Belmont, CA, USA, 2001.Schierle, G. Architectural Structures Excerpts; University of Southern California: Los Angeles, CA, USA, 2006.Chen, W.F.; Duan, L. Bridge Engineering Handbook; CRC Press: Boca Raton, FL, USA, 2000.Pollock, T.I.; Panagiotopoulou, O.; Hocking, D.P.; Evans, A.R. Taking a stab at modelling canine tooth biomechanics in mammalian carnivores with beam theory and finite-element analysis. R. Soc. Open Sci. 2022, 9, 220701.Le Huec, J.C.; Droulout, T.; Boue, L.; Dejour, E.; Ramos-Pascual, S.; Bourret, S. A novel device with pedicular anchorage provides better biomechanical properties than balloon kyphoplasty for the treatment of vertebral compression fractures. J. Exp. Orthop. 2023, 10, 71.Minutolo, V.; Esposito, L.; Sacco, E.; Fraldi, M. Designing stress for optimizing and toughening truss-like structures. Meccanica 2020, 55, 1603–1622.Medina, L.; Gilat, R.; Ilic, B.; Krylov, S. Experimental investigation of the snap-through buckling of electrostatically actuated initially curved pre-stressed micro beams. Sens. Actuators A Phys. 2014, 220, 323–332.Patil, A.S.; Arnold, E. Characterization of standard structural CFRP beam shapes for UAS VHF antenna applications. In Proceedings of the AIAA Scitech 2021 Forum, Virtual, 11–15 & 19–21 January 2021; p. 1803.Allen, E.; Zalewski, W. Form and Forces; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2010.Gozluklu, B.; Uyar, I.; Coker, D. Intersonic delamination in curved thick composite laminates under quasi-static loading. Mech. Mater. 2015, 80, 163–182.Peterson, D.R.; Bronzino, J.D. Biomechanics: Principles and Applications; CRC Press: Boca Raton, FL, USA, 2008.Winkelstein, B.A. Orthopaedic Biomechanics; CRC Press: Boca Raton, FL, USA, 2013.Timoshenko, S.P.; Goodier, J.N. Theory of Elasticity; McGraw-Hill Inc.: New York, NY, USA, 1970.Eason, G. The elastic-plastic bending of a compressible curved bar. Appl. Sci. Res. 1960, 9, 53–63.Lekhnitskii, S.G.; Tsai, S.W.; Cheron, T. Anisotropic Plates; Gordon and Breach: New York, NY, USA, 1968.Kedward, K.T.; Wilson, R.S.; McLean, S.K. Flexure of simply curved composite shapes. Composites 1989, 20, 527–536.Bagci, C. Exact Elasticity Solutions for Stresses and Deflections in Curved Beams and Rings of Exponential and T-Sections. J. Mech. Des. 1993, 115, 346–358.Ugural, A.C.; Fenster, S.K. Advanced Mechanics of Materials and Applied Elasticity; Pearson Education Inc.: Boston, MA, USA, 2020.Sloboda, A.; Honarmandi, P. Generalized Elasticity Method for Curved Beam Stress Analysis: Analytical and Numerical Comparisons for a Lifting Hook. Mech. Based Des. Struct. Mach. 2007, 35, 319–332.Black, W.E. Discussion: Stresses in Curved Beams—A Tabular Method of Solution Based on Winkler’s Theory. J. Appl. Mech. 1953, 20, 444–445.Bleich, H. Die Spannungsverteilung in den Gurtungen gekrümmter Stäbe mit T- und I-formigen Querschnitt. Der. Stahlbau, Beil. Zur Z. Die Bautech. 1933, 6, 3–6.Anderson, C.G. Flexural Stresses in Curved Beams of I- and Box Sections. Proc. Inst. Mech. Eng. 1950, 163, 295–306.Cook, R.D. Circumferential Stresses in Curved Beams. J. Appl. Mech. 1992, 59, 224–225.Young, W.C.; Cook, R.D. Radial Stress Formula for Curved Beams. J. Vib. Acoust. 1989, 111, 491–492.Wang, T.S. Shear Stresses in Curved Beams. Mach. Des. 1967, 39, 175–178Birger, I.A.; Panovko, J.G. Strength-Stability-Vibrations; Machinery Publishing House: Moscow, Russia, 1968.Oden, J.; Ripperger, E. Mechanics of Elastic Structures; McGraw-Hill: New York, NY, USA, 1981; pp. 101–107.Liu, H. Advanced Strength of Materials; Advance Education Publishing House: Beijing, China, 1985. (In Chinese)Yu, A.; Nie, G. Explicit solutions for shearing and radial stresses in curved beams. Mech. Res. Commun. 2005, 32, 323–331.Iandiorio, C.; Salvini, P. An Engineering Theory of Thick Curved Beams Loaded In-Plane and Out-of-Plane: 3D Stress Analysis. Eur. J. Mech./A Solids 2022, 92, 104484.Wang, Y.P.; Lee, C.L.; Huang, S.C. Inelastic Stress Analysis of Curved Beams with Bending and Shear Coupling. In Proceedings of the World Congress on Civil, Structural, and Environmental Engineering, CSEE’16, Prague, Czech Republic, 30–31 March 2016.Nahvi, H. Pure Bending and Tangential Stresses in Curved Beams of Trapezoidal and Circular Sections. J. Mech. Behav. Mater. 2007, 18, 123–132.Sayyad, A.S.; Ghugal, Y.M. Bending, Buckling and Free Vibration of Laminated Composite and Sandwich Beams: A Critical Review of Literature. Compos. Struct. 2017, 171, 486–504.Hajianmaleki, M.; Qatu, M.S. Vibrations of Straight and Curved Composite Beams: A Review. Compos. Struct. 2013, 100, 218–232.Li, W.; Ma, H.; Gao, W. Geometrically Exact Beam Element with Rational Shear Stress Distribution for Nonlinear Analysis of FG Curved Beams. Thin-Walled Struct. 2021, 164, 107823.Ferradi, M.K.; Cespedes, X. A Curved Beam Model with the Asymptotic Expansion Method. Eng. Struct. 2021, 241, 112494.Gao, Y.; Wang, M.Z.; Zhao, B.S. The Refined Theory of Rectangular Curved Beams. Acta Mech. 2006, 189, 141–150.Thurnherr, C.; Groh, R.M.J.; Ermanni, P.; Weaver, P.M. Higher-Order Beam Model for Stress Predictions in Curved Beams Made from Anisotropic Materials. Inter. J. Solids Struct. 2016, 97–98, 16–28.Ghuku, S.; Saha, K. A review on stress and deformation analysis of curved beams under large deflection. Int. J. Eng. Technol. 2017, 11, 13–39.Cheung, K.; Sorensen, H. Effect of loads on radial stress in curved beams. Soc. Wood Sci. Technol. 1983, 15, 263–275.Hassan, I. Experimental and analytical study of bending stresses and deflections in curved beam made of laminated composite material. AL-Khwarizmi Eng. J. 2014, 10, 21–32.Prasad, S.; Subramanian, R.; Krishna, S.; Prashanth, S. Experimental stress analysis of curved beams using strain gauges. Int. J. Latest Technol. Eng. Manag. Appl. Sci. 2016, V, 1–6.Ahuett-Garza, H.; Chaides, O.; Garcia, P.N.; Urbina, P. Studies about the use of semicircular beams as hinges in large deflection planar compliant mechanisms. Precis. Eng. 2014, 38, 711–727.Yanze, L.; Ke, Z.; Huaitao, S.; Songhua, L.; Xiaochen, Z. Theoretical and Experimental Analysis of Thin-Walled Curved Rectangular Box Beam under In-Plane Bending. Scanning 2021, 2021, 8867142.Pai, P.F.; Anderson, T.J.; Wheater, E.A. Large-deformation tests and total-Lagrangian finite-element analyses of flexible beams. Int. J. Solids Struct. 2000, 37, 2951–2980.Aşık, M.Z.; Tezcan, S. A mathematical model for the behavior of laminated glass beams. Comput. Struct. 2005, 83, 1742–1753.Angel, G.; Haritos, G.; Chrysanthou, A.; Voloshin, V. Chord line force versus displacement for thin shallow arc pre-curved bimetallic strip. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 2015, 229, 116–124Boresi, A.P.; Schmidt, R.J. Advanced Mechanics of Materials; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2003Fazlali, M.R.; Arghavani, J.; Eskandari, M. An Analytical Study on the Elastic-Plastic Pure Bending of a Linear Kinematic Hardening Curved Beam. Int. J. Mech. Sci. 2018, 144, 274–282.Seely, F.B.; Smith, J.O. Advanced Mechanics of Materials; John Wiley & Sons, Inc.: New York, NY, USA, 1952.Cook, R.D.; Young, W.C. Advanced Mechanics of Materials; Macmillan: New York, NY, USA, 1985.Tsao, C.H. Radial Stresses for Curved Beams. J. Vib. Acoust. 1986, 108, 107–108.Ortiz Berrocal, L. Resistencia de Materiales; McGraw-Hill: Madrid, Spain, 1990.Ansys User’s Manual: Theory Reference; 2023R1; The University of Texas Houston: Houston, TX, USA, 2023.http://purl.org/coar/resource_type/c_2df8fbb1ORIGINALmaterials-17-05982.pdfmaterials-17-05982.pdfapplication/pdf14267979https://repositorio.utb.edu.co/bitstream/20.500.12585/12952/1/materials-17-05982.pdfc5c90530d810b22376701231ec5edbe2MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8701https://repositorio.utb.edu.co/bitstream/20.500.12585/12952/2/license_rdf42fd4ad1e89814f5e4a476b409eb708cMD52LICENSElicense.txtlicense.txttext/plain; charset=utf-83182https://repositorio.utb.edu.co/bitstream/20.500.12585/12952/3/license.txte20ad307a1c5f3f25af9304a7a7c86b6MD53TEXTmaterials-17-05982.pdf.txtmaterials-17-05982.pdf.txtExtracted texttext/plain53935https://repositorio.utb.edu.co/bitstream/20.500.12585/12952/4/materials-17-05982.pdf.txtb8ad66064a9821d31ebfc02c62d1f62fMD54THUMBNAILmaterials-17-05982.pdf.jpgmaterials-17-05982.pdf.jpgGenerated Thumbnailimage/jpeg7714https://repositorio.utb.edu.co/bitstream/20.500.12585/12952/5/materials-17-05982.pdf.jpgf15bd18ce857387ddce567d8401cabf9MD5520.500.12585/12952oai:repositorio.utb.edu.co:20.500.12585/129522024-12-07 00:15:42.241Repositorio Institucional UTBrepositorioutb@utb.edu.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 |