Shape Reconstruction of a Timoshenko Beam under the Geometric Nonlinearity Condition
Publication: Journal of Engineering Mechanics
Volume 149, Issue 6
Abstract
Shape sensing, which is the real-time monitoring of deformed shapes using discrete surface strain, is a fundamental approach to ensure structural safety, reliability, and affordability. Large deformation shape sensing is obviously more important because large deformations can result in structural damage and failure. Nevertheless, there are few effective methods for the shape sensing of large deformations. Based on Timoshenko beam theory, this paper establishes a new method, called analogy stiffness upgrading (ASU), to reconstruct nonlinear deformation. In this method, the inverse finite element method (iFEM) is used to predict the initial displacement field and compute the analogy stiffness matrix. Then, the analogy stiffness matrix is upgraded by using coordinate transformation from a co-rotational procedure. Through iterative computation, the real displacement field is finally obtained when the rotation angle calculated from the input strain data is the same as the integral result from the section strain data. Numerical examples and model tests are carried out to verify the ASU method. It is evident from the results that the ASU method can predict largely deformed shapes of beam structures with superior precision.
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Data Availability Statement
All data, models, or code that support the findings of this study are available from the corresponding author on reasonable request.
Acknowledgments
This research work is jointly supported by the National Natural Science Foundation of China (Grant Nos. 52008236 and 52078284), Guangdong Basic and Applied Basic Research Foundation (2022A1515010812 and 2021A1515011770), and Shantou University Scientific Research Foundation (Grant Nos. NTF19039 and NTF18012). These grants are greatly appreciated.
References
Alioli, M., P. Masarati, M. Morandini, G. L. Ghiringhelli, T. Carpenter, and R. Albertani. 2015. “Nonlinear membrane inverse finite elements.” AIP Conf. Proc. 1648 (1): 120002. https://doi.org/10.1063/1.4912419.
Battini, J.-M., and C. Pacoste. 2002. “Co-rotational beam elements with warping effects in instability problems.” Comput. Methods Appl. Mech. Eng. 191 (17–18): 1755–1789. https://doi.org/10.1016/S0045-7825(01)00352-8.
Bernardini, G., R. Porcelli, J. Serafini, and P. Masarati. 2018. “Rotor blade shape reconstruction from strain measurements.” Aerosp. Sci. Technol. 79 (Aug): 580–587. https://doi.org/10.1016/j.ast.2018.06.012.
Crisfield, M. A. 1990. “A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements.” Comput. Methods Appl. Mech. Eng. 81 (2): 131–150. https://doi.org/10.1016/0045-7825(90)90106-V.
Deng, H. X., H. C. Zhang, J. Wang, J. Zhang, M. C. Ma, and X. Zhong. 2019. “Modal learning displacement-strain transformation.” Rev. Sci. Instrum. 90 (7): 075113. https://doi.org/10.1063/1.5100905.
Dewangan, H. C., and S. K. Panda. 2022. “Large deformation effect on dynamic deflection responses of cutout-borne composite shell panel: An experimental validation.” J. Eng. Mech. 148 (8): 04022042. https://doi.org/10.1061/(ASCE)EM.1943-7889.0002129.
Esposito, M., and M. Gherlone. 2020. “Composite wing box deformed-shape reconstruction based on measured strains: Optimization and comparison of existing approaches.” Aerosp. Sci. Technol. 99 (Apr): 105758. https://doi.org/10.1016/j.ast.2020.105758.
Felippa, C. A., and B. Haugen. 2005. “A unified formulation of small-strain corotational finite elements: I. Theory.” Comput. Methods Appl. Mech. Eng. 194 (21–24): 2285–2335. https://doi.org/10.1016/j.cma.2004.07.035.
Foss, G. C., and E. D. Haugse. 1995. “Using modal test results to develop strain to displacement transformations.” In Vol. 2460 of Proc., 13th Int. Modal Analysis Conf., 112. Bethel, CT: Society for Experimental Mechanics.
Garg, P., F. Moreu, A. Ozdagli, M. R. Taha, and D. Mascareñas. 2019. “Noncontact dynamic displacement measurement of structures using a moving laser Doppler vibrometer.” J. Bridge Eng. 24 (9): 04019089. https://doi.org/10.1061/(ASCE)BE.1943-5592.0001472.
Gherlone, M., P. Cerracchio, and M. Mattone. 2018. “Shape sensing methods: Review and experimental comparison on a wing-shaped plate.” Prog. Aerosp. Sci. 99 (May): 14–26. https://doi.org/10.1016/j.paerosci.2018.04.001.
Gherlone, M., P. Cerracchio, M. Mattone, M. Di Sciuva, and A. Tessler. 2012. “Shape sensing of 3D frame structures using an inverse finite element method.” Int. J. Solids Struct. 49 (22): 3100–3112. https://doi.org/10.1016/j.ijsolstr.2012.06.009.
Joshi, S., and S. M. Harle. 2017. “Linear variable differential transducer (LVDT) and its applications in civil engineering.” Int. J. Transp. Eng. Technol. 3 (4): 62–66. https://doi.org/10.11648/j.ijtet.20170304.13.
Kefal, A. 2019. “An efficient curved inverse-shell element for shape sensing and structural health monitoring of cylindrical marine structures.” Ocean Eng. 188 (Sep): 106262. https://doi.org/10.1016/j.oceaneng.2019.106262.
Kefal, A., E. Oterkus, A. Tessler, and J. L. Spangler. 2016. “A quadrilateral inverse-shell element with drilling degrees of freedom for shape sensing and structural health monitoring.” Eng. Sci. Technol. 19 (3): 1299–1313. https://doi.org/10.1016/j.jestch.2016.03.006.
Ko, W. L., L. Richards, and V. T. Fleischer. 2009. Applications of Ko displacement theory to the deformed shape predictions of the doubly-tapered Ikhana wing. Edwards, CA: NASA Dryden Flight Research Center.
Ko, W. L., W. L. Richards, and V. T. Tran. 2007. Displacement theories for in-flight deformed shape predictions of aerospace structures. Edwards, CA: NASA Dryden Flight Research Center.
Kohut, P., K. Holak, T. Uhl, Ł. Ortyl, T. Owerko, P. Kuras, and R. Kocierz. 2013. “Monitoring of a civil structure’s state based on noncontact measurements.” Struct. Health Monit. 12 (5–6): 411–429. https://doi.org/10.1177/1475921713487397.
Li, M., A. Kefal, E. Oterkus, and S. Oterkus. 2020. “Structural health monitoring of an offshore wind turbine tower using iFEM methodology.” Ocean Eng. 204 (May): 107291. https://doi.org/10.1016/j.oceaneng.2020.107291.
Li, Z., B. Izzuddin, and L. Vu-Quoc. 2008. “A 9-node co-rotational quadrilateral shell element.” Comput. Mech. 42 (6): 873–884. https://doi.org/10.1007/s00466-008-0289-8.
Liu, H., S. Zhang, A. A. Coulibaly, J. Cheng, and M. J. DeJong. 2021. “Monitoring reinforced concrete cracking behavior under uniaxial tension using distributed fiber-optic sensing technology.” J. Struct. Eng. 147 (12): 04021212. https://doi.org/10.1061/(ASCE)ST.1943-541X.0003191.
Lively, P. S., M. J. Atalla, and N. W. Hagood. 2001. “Investigation of filtering techniques applied to the dynamic shape estimation problem.” Smart Mater. Struct. 10 (2): 264. https://doi.org/10.1088/0964-1726/10/2/311.
Ma, Q., and M. Solís. 2019. “Multiple damage identification in beams from full-field digital photogrammetry.” J. Eng. Mech. 145 (8): 04019054. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001629.
Ma, Z., J. Choi, and H. Sohn. 2022. “Real-time structural displacement estimation by fusing asynchronous acceleration and computer vision measurements.” Comput.-Aided Civ. Infrastruct. Eng. 37 (6): 688–703. https://doi.org/10.1111/mice.12767.
Morgese, M., Y. Ying, T. Taylor, and F. Ansari. 2022. “Method and theory for conversion of distributed fiber-optic strains to crack opening displacements.” J. Eng. Mech. 148 (12): 04022072. https://doi.org/10.1061/(ASCE)EM.1943-7889.0002168.
Nasimi, R., and F. Moreu. 2021. “Development and implementation of a laser–camera–UAV System to measure total dynamic transverse displacement.” J. Eng. Mech. 147 (8): 04021045. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001939.
Paczkowski, K., and H. Riggs. 2007. “An inverse finite element strategy to recover full-field, large displacements from strain measurements.” In Proc., Int. Conf. on Offshore Mechanics and Arctic Engineering. New York: Offshore, and Arctic Engineering Division of ASME. https://doi.org/10.1115/OMAE2007-29730.
Pak, C. G. 2016. “Wing shape sensing from measured strain.” AIAA. J. 54 (3): 1068–1077. https://doi.org/10.2514/1.J053986.
Ren, L., T. Jiang, Z.-G. Jia, D.-S. Li, C.-L. Yuan, and H.-N. Li. 2018. “Pipeline corrosion and leakage monitoring based on the distributed optical fiber sensing technology.” Measurement 122 (Jul): 57–65. https://doi.org/10.1016/j.measurement.2018.03.018.
Shen, S., Z. Wu, C. Yang, C. Wan, Y. Tang, and G. Wu. 2010. “An improved conjugated beam method for deformation monitoring with a distributed sensitive fiber optic sensor.” Struct. Health Monit. 9 (4): 361–378. https://doi.org/10.1177/1475921710361326.
Tessler, A., R. Roy, M. Esposito, C. Surace, and M. Gherlone. 2018. “Shape sensing of plate and shell structures undergoing large displacements using the inverse finite element method.” Shock. Vib. 2018: 88–95. https://doi.org/10.1155/2018/8076085.
Tessler, A., and J. L. Spangler. 2003. A variational principle for reconstruction of elastic deformations in shear deformable plates and shells. Hampton, VA: National Aeronautics and Space Administration, Langley Research Center.
Tessler, A., and J. L. Spangler. 2005. “A least-squares variational method for full-field reconstruction of elastic deformations in shear-deformable plates and shells.” Comput. Methods Appl. Mech. Eng. 194 (2–5): 327–339. https://doi.org/10.1016/j.cma.2004.03.015.
Ye, J., G. Fu, and U. P. Poudel. 2011. “Edge-based close-range digital photogrammetry for structural deformation measurement.” J. Eng. Mech. 137 (7): 475–483. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000251.
You, R., L. Ren, C. Yuan, and G. Song. 2021. “Two-dimensional deformation estimation of beam-like structures using inverse finite-element method: Theoretical study and experimental validation.” J. Eng. Mech. 147 (5): 04021019. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001917.
Zhao, F., A. Kefal, and H. Bao. 2022. “Nonlinear deformation monitoring of elastic beams based on isogeometric iFEM approach.” Int. J. Non-Linear Mech. 147 (Dec): 104229. https://doi.org/10.1016/j.ijnonlinmec.2022.104229.
Zhou, Y., and L. Sun. 2019. “Insights into temperature effects on structural deformation of a cable-stayed bridge based on structural health monitoring.” Struct. Health Monit. 18 (3): 778–791. https://doi.org/10.1177/1475921718773954.
Zhu, C., Z.-D. Xu, H. Lu, and Y. Lu. 2022. “Evaluation of cross-sectional deformation in pipes using reflection of fundamental guided-waves.” J. Eng. Mech. 148 (5): 04022016. https://doi.org/10.1061/(ASCE)EM.1943-7889.0002095.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 149 • Issue 6 • June 2023
Copyright
© 2023 American Society of Civil Engineers.
History
Received: Dec 1, 2022
Accepted: Jan 26, 2023
Published online: Mar 29, 2023
Published in print: Jun 1, 2023
Discussion open until: Aug 29, 2023
ASCE Technical Topics:
- Beams
- Continuum mechanics
- Deformation (mechanics)
- Design (by type)
- Engineering fundamentals
- Engineering mechanics
- Finite element method
- Geometrics
- Highway and road design
- Material mechanics
- Materials engineering
- Measurement (by type)
- Methodology (by type)
- Numerical methods
- Sensors and sensing
- Solid mechanics
- Stiffening
- Strain
- Structural behavior
- Structural design
- Structural engineering
- Structural mechanics
- Structural members
- Structural safety
- Structural systems
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Cited by
- Tao Jiang, Jing-wen Zhu, Ming-zhao Xian, Dong-sheng Li, Shape Reconstruction Method for Monitoring Large Deformed Beam Structures, Journal of Engineering Mechanics, 10.1061/JENMDT.EMENG-7530, 150, 8, (2024).