Skip to main content
SpringerLink
Log in

Shape sensing modeling of Timoshenko beam based on the strain gradient theory and iFEM method

基于应变梯度理论和逆有限元法的Timoshenko梁形状传感建模

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

The geometrically nonlinear deformation of the large scale structure seriously endangers the structural system safety. Thus, it is of great significance to real-time monitor the structural deformation in service. However, the current inverse finite element method (iFEM), which is presented based on the linear elastic theory, is not suitable for nonlinear deformation. This paper proposes a nonlinear iFEM for establishing the shape sensing model. Initially, the kinematics and kinetics of the strain-gradient Timoshenko beam model are presented, and the governing equations for geometrically nonlinear behavior are formulated. Then, the analytical solution of the rotation function is presented and the nonlinear shape sensing model is established. Therewith, isogeometric analysis (IGA) approach is employed to construct the interpolation shape functions. Due to displacement functions expressed in terms of rotations, the “shear locking” problem can be effectively avoided. Subsequently, the experimental rotation functions are deduced using discrete surface strain measurements and the rotation transformation is established between Cartesian and curvilinear coordinate systems. Finally, a cantilevered beam is used as a case study to compare the reconstructed with theoretical displacements. The numerical results demonstrate the excellent performance of the proposed formulation, where the reconstructed errors are less than 2.5% for both concentrated and distributed loads.

摘要

大型结构的几何非线性变形严重威胁着结构体系的安全. 因此, 对在役结构变形进行实时监测具有重要意义. 然而, 目前基于线 弹性理论提出的逆有限元法(iFEM)并不适用于非线性变形, 为此, 提出了一种非线性逆有限元法来建立结构形状感知模型. 首先给出 了应变梯度Timoshenko梁模型的运动学和动力学变量, 并建立了其几何非线性变形控制方程. 然后, 推导出了转角函数的解析解, 并建 立了非线性形状感知模型. 其中, 采用等几何分析(IGA)方法构造插值形状函数. 由于位移函数以旋转形式表示, 所以可以有效地避免 “剪切锁定”问题. 随后, 利用离散表面应变测量值推导出了实际转角函数, 并建立了转角自由度在直角坐标系与曲线坐标系之间转换关 系. 最后, 以一悬臂梁为例, 将重构位移与理论位移进行了比较. 数值结果表明, 不论是集中荷载还是分布荷载, 所提的非线性逆有限元 法均具有良好的重构性能, 重构误差都小于2.5%.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. H. Zhu, Z. Du, and Y. Tang, Numerical study on the displacement reconstruction of subsea pipelines using the improved inverse finite element method, Ocean Eng. 248, 110763 (2022).

    Article  Google Scholar 

  2. V. Giurgiutiu, Structural health monitoring (SHM) of aerospace composites, Polym. Compos. Aeros. Ind. 2015, 491 (2020).

    Google Scholar 

  3. M. Sheykhi, A. Eskandari, D. Ghafari, R. Ahmadi Arpanahi, B. Mohammadi, and S. Hosseini Hashemi, Investigation of fluid viscosity and density on vibration of nano beam submerged in fluid considering nonlocal elasticity theory, Alexandria Eng. J. 65, 607 (2023).

    Article  Google Scholar 

  4. A. Kefal, C. Diyaroglu, M. Yildiz, and E. Oterkus, Coupling of peridynamics and inverse finite element method for shape sensing and crack propagation monitoring of plate structures, Comput. Methods Appl. Mech. Eng. 391, 114520 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  5. W. L. Ko, W. L. Richards, and V. T. Fleischer, Applications of Ko displacement theory to the deformed shape predictions of the doubly-tapered Ikhana Wing, NASA/TP-2009-214652, 2009.

  6. W. L. Ko, W. L. Richards, and V. T. Fleischer, Displacement theories for In-flight deformed shape predictions of aerospace structure, NASA/TP-2007-214612, 2007.

  7. W. L. Ko, and W. L. Richards, Further development of Ko displacement theory for deformed shape predictions of nonuniform aerospace structures, NASA-TP-2009-214643, 2009.

  8. C. V. Jutte, W. L. Ko, C. A. Stephens, J. A. Bakalyar, and W. L. Richards, Deformed shape calculation of a full-scale wing using fiber optic strain data from a ground loads test, NASA/TP-2011-215975, 2011.

  9. B. Smoke, and A. Baz, Monitoring the bending and twist of morphing structures, in: Proceedings of the SPIE6932, Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace System, San Diego, 2008.

    Google Scholar 

  10. G. Foss, and E. Haugse, Using modal test results to develop strain to displacements transformations, in: Proceeding of the 13th International Conference on Modal Analysis, Nashville, 1995.

  11. A. C. Pisoni, C. Santolini, and D. E. Hauf, Displacements in a vibrating body by strain gauge measurements, in: Proceedings of the 13th International Conference on Modal Analysis, Nashville, 1995.

  12. M. A. Davis, A. D. Kersey, J. Sirkis, and E. J. Friebele, Shape and vibration mode sensing using a fiber optic Bragg grating array, Smart Mater. Struct. 5, 759 (1996).

    Article  ADS  CAS  Google Scholar 

  13. N. S. Kim, and N. S. Cho, Estimating deflection of a simple beam model using fiber optic bragg-grating sensors, Exp. Mech. 44, 433 (2004).

    Article  Google Scholar 

  14. L. Moreira, and C. G. Soares, Neural network model for estimation of hull bending moment and shear force of ships in waves, Ocean Eng. 206, 107347 (2020).

    Article  Google Scholar 

  15. H. Nguyen, T. Tran, Y. Wang, and Z. Wang, Three-dimensional shape reconstruction from single-shot speckle image using deep convolutional neural networks, Opt. Lasers Eng. 143, 106639 (2021).

    Article  Google Scholar 

  16. K. Xu, and E. Darve, Solving inverse problems in stochastic models using deep neural networks and adversarial training, Comput. Methods Appl. Mech. Eng. 384, 113976 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  17. A. Tessler, and J. L. Spangler, A least-squares variational method for full-field reconstruction of elastic deformations in shear-deformable plates and shells, Comput. Methods Appl. Mech. Eng. 194, 327 (2005).

    Article  ADS  Google Scholar 

  18. M. Gherlone, P. Cerracchio, M. Mattone, M. Di Sciuva, and A. Tessler, Shape sensing of 3D frame structures using an inverse finite element method, Int. J. Solids Struct. 49, 3100 (2012).

    Article  Google Scholar 

  19. M. Gherlone, P. Cerracchio, and M. Mattone, Shape sensing methods: Review and experimental comparison on a wing-shaped plate, Prog. Aerosp. Sci. 99, 14 (2018).

    Article  Google Scholar 

  20. M. Esposito, and M. Gherlone, Composite wing box deformed-shape reconstruction based on measured strains: Optimization and comparison of existing approaches, Aerosp. Sci. Tech. 99, 105758 (2020).

    Article  Google Scholar 

  21. A. Tessler, and J. L. Spangler, in Inverse FEM for full-field reconstruction of elastic deformations in shear deformable plates and shells: Proceedings of 2nd European Workshop on Structural Health Monitoring, Munich, 2004.

  22. A. Kefal, E. Oterkus, A. Tessler, and J. L. Spangler, A quadrilateral inverse-shell element with drilling degrees of freedom for shape sensing and structural health monitoring, Eng. Sci. Tech. Int. J. 19, 1299 (2016).

    Google Scholar 

  23. A. Kefal, An efficient curved inverse-shell element for shape sensing and structural health monitoring of cylindrical marine structures, Ocean Eng. 188, 106262 (2019).

    Article  Google Scholar 

  24. D. Oboe, L. Colombo, C. Sbarufatti, and M. Giglio, Shape sensing of a complex aeronautical structure with inverse finite element method, Sensors 21, 1388 (2021).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  25. A. Kefal, J. B. Mayang, E. Oterkus, and M. Yildiz, Three dimensional shape and stress monitoring of bulk carriers based on iFEM methodology, Ocean Eng. 147, 256 (2018).

    Article  Google Scholar 

  26. A. Kefal, and E. Oterkus, Displacement and stress monitoring of a chemical tanker based on inverse finite element method, Ocean Eng. 112, 33 (2016).

    Article  Google Scholar 

  27. H. Zhu, Z. Du, and Y. Tang, Automatic free span assessment for subsea pipelines using static strain data, Ocean Eng. 263, 112413 (2022).

    Article  Google Scholar 

  28. A. Kefal, and E. Oterkus, Isogeometric iFEM analysis of thin shell structures, Sensors 20, 2685 (2020).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  29. F. Zhao, L. Xu, H. Bao, and J. Du, Shape sensing of variable cross-section beam using the inverse finite element method and isogeometric analysis, Measurement 158, 107656 (2020).

    Article  Google Scholar 

  30. A. Kefal, and E. Oterkus, in Shape sensing of aerospace structures by coupling of isogeometric analysis and inverse finite element method: Proceedings of 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Grapevine, 2017.

  31. K. Chen, K. Cao, G. Gao, and H. Bao, Shape sensing of Timoshenko beam subjected to complex multi-node loads using isogeometric analysis, Measurement 184, 109958 (2021).

    Article  Google Scholar 

  32. A. Tessler, M. Di Sciuva, and M. Gherlone, A consistent refinement of first-order shear deformation theory for laminated composite and sandwich plates using improved zigzag kinematics, J. Mech. Mater. Struct. 5, 341 (2010).

    Article  Google Scholar 

  33. P. Cerracchio, M. Gherlone, M. Di Sciuva, and A. Tessler, A novel approach for displacement and stress monitoring of sandwich structures based on the inverse finite element method, Compos. Struct. 127, 69 (2015).

    Article  Google Scholar 

  34. A. Kefal, A. Tessler, and E. Oterkus, An enhanced inverse finite element method for displacement and stress monitoring of multilayered composite and sandwich structures, Compos. Struct. 179, 514 (2017).

    Article  Google Scholar 

  35. F. Zhao, H. Bao, J. Liu, and K. Li, Shape sensing of multilayered composite and sandwich beams based on Refined Zigzag Theory and inverse finite element method, Compos. Struct. 261, 113321 (2021).

    Article  Google Scholar 

  36. A. Kefal, and M. Yildiz, Modeling of sensor placement strategy for shape sensing and structural health monitoring of a wing-shaped sandwich panel using inverse finite element method, Sensors 17, 2775 (2017).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  37. M. A. Abdollahzadeh, I. E. Tabrizi, A. Kefal, and M. Yildiz, A combined experimental/numerical study on deformation sensing of sandwich structures through inverse analysis of pre-extrapolated strain measurements, Measurement 185, 110031 (2021).

    Article  Google Scholar 

  38. A. Kefal, I. E. Tabrizi, M. Yildiz, and A. Tessler, A smoothed iFEM approach for efficient shape-sensing applications: Numerical and experimental validation on composite structures, Mech. Syst. Signal Process. 152, 107486 (2021).

    Article  Google Scholar 

  39. S. Niu, Y. Zhao, and H. Bao, Shape sensing of plate structures through coupling inverse finite element method and scaled boundary element analysis, Measurement 190, 110676 (2022).

    Article  Google Scholar 

  40. D. Oboe, C. Sbarufatti, and M. Giglio, Physics-based strain pre-extrapolation technique for inverse Finite Element Method, Mech. Syst. Signal Process. 177, 109167 (2022).

    Article  Google Scholar 

  41. F. Zhao, H. Bao, and J. Du, A real-time deformation displacement measurement method for Timoshenko beams with multiple singularities, IEEE Trans. Instrum. Meas. 70, 1 (2021).

    Google Scholar 

  42. A. Tessler, R. Roy, M. Esposito, C. Surace, and M. Gherlone, Shape sensing of plate and shell structures undergoing large displacements using the inverse finite element method, Shock Vib. 2018, 1 (2018).

    Article  Google Scholar 

  43. M. A. Abdollahzadeh, H. Q. Ali, M. Yildiz, and A. Kefal, Experimental and numerical investigation on large deformation reconstruction of thin laminated composite structures using inverse finite element method, Thin-Walled Struct. 178, 109485 (2022).

    Article  Google Scholar 

  44. F. Zhao, A. Kefal, and H. Bao, Nonlinear deformation monitoring of elastic beams based on isogeometric iFEM approach, Int. J. NonLinear Mech. 147, 104229 (2022).

    Article  ADS  Google Scholar 

  45. A. Beheshti, Large deformation analysis of strain-gradient elastic beams, Comput. Struct. 177, 162 (2016).

    Article  Google Scholar 

  46. F. Dadgar-Rad, and S. Sahraee, Large deformation analysis of fully incompressible hyperelastic curved beams, Appl. Math. Model. 93, 89 (2021).

    Article  MathSciNet  Google Scholar 

  47. M. Maleki, S. A. M. Tonekaboni, and S. Abbasbandy, A homotopy analysis solution to large deformation of beams under static arbitrary distributed load, Appl. Math. Model. 38, 355 (2014).

    Article  MathSciNet  Google Scholar 

  48. H. Ghaffarzadeh, and A. Nikkar, Explicit solution to the large deformation of a cantilever beam under point load at the free tip using the variational iteration method-II, J. Mech. Sci. Technol. 27, 3433 (2013).

    Article  Google Scholar 

  49. S. Hosseini-Hashemi, R. A. Arpanahi, S. Rahmanian, and A. Ahmadi-Savadkoohi, Free vibration analysis of nano-plate in viscous fluid medium using nonlocal elasticity, Eur. J. Mech.-A Solids 74, 440 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  50. R. A. Arpanahi, S. Hosseini-Hashemi, S. Rahmanian, S. H. Hashemi, and A. Ahmadi-Savadkoohi, Nonlocal surface energy effect on free vibration behavior of nanoplates submerged in incompressible fluid, Thin-Walled Struct. 143, 106212 (2019).

    Article  Google Scholar 

  51. R. A. Arpanahi, A. Eskandari, S. Hosseini-Hashemi, M. Taherkhani, and S. H. Hashemi, Surface energy effect on free vibration characteristics of nano-plate submerged in viscous fluid, J. Vib. Eng. Technol. (2023).

  52. T. Beléndez, C. Neipp, and A. Beléndez, Large and small deflections of a cantilever beam, Eur. J. Phys. 23, 371 (2002).

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51775401, and 51675398).

Author information

Authors and Affiliations

  1. School of Electronic Engineering, Xidian University, Xi’an, 710071, China

    Feifei Zhao  (赵飞飞), Yanhao Guo  (郭彦浩), Hong Bao  (保宏) & Wei Wang  (王伟)

  2. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, 710072, China

    Feng Zhang  (张峰)

Authors
  1. Feifei Zhao  (赵飞飞)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yanhao Guo  (郭彦浩)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hong Bao  (保宏)
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wei Wang  (王伟)
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Feng Zhang  (张峰)
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Author contributions Feifei Zhao performed the majority of work on the paper, reviewing the inverse finite method, proposing non-iFEM algorithm, and writing the article. Yanhao Guo and Hong Bao performed the formal analysis and experimental validation. Wei Wang established the finite element model and performed simulation analyses. Feng Zhang gave some advice about the paper structure and checked the grammar structure for this paper.

Corresponding author

Correspondence to Feifei Zhao  (赵飞飞).

Ethics declarations

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, F., Guo, Y., Bao, H. et al. Shape sensing modeling of Timoshenko beam based on the strain gradient theory and iFEM method. Acta Mech. Sin. 39, 423039 (2023). https://doi.org/10.1007/s10409-023-23039-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10409-023-23039-x

Keywords

Search

Navigation