Multi-harmonic balance method in structural element ratcheting problems
DOI:
https://doi.org/10.7242/1999-6691/2025.18.3.21Keywords:
ratcheting, cyclic loads, shakedown, pressure vessel, thermal cyclic loads, plastic strain, numerical simulation, ABAQUSAbstract
The ratcheting theory originated in the 1960s. In mechanics of deformable bodies, it is used to assess the strength of structural elements (gas turbines, high-temperature reactors, etc.) operating under intense cyclic thermal stresses. This study aims to investigate possible modes of elastic-plastic deformation in pressure vessels under cyclically varying temperature conditions, assuming an elastic-ideal plastic material behavior model. Due to material hardening, analytical methods are not sufficient to solve this problem, and numerical methods must be used. The number of calculation cycles significantly affects a process's result: a large number of these cycles can create modeling difficulties due to limited time and computational resources. Therefore, the Direct Cyclic Method (DCM) has become increasingly popular in engineering practice for solving this type of problems. This method is an application of the multi-harmonic balance technique to the problems of quasistatic cyclic loading of viscoelastic-plastic materials. The validity of the DCM method is evaluated by comparing it with the method of direct nonlinear analysis of loading history when assessing the adaptability of vessels under cyclic thermal stresses. The computational efficiency of the DCM was investigated for varying finite element mesh sizes and loading cycles. It is shown that the limitations of the DCM become apparent in cases where a stable periodic solution does not exist.
Downloads
References
GOST R 59115.10-2021. Obosnovaniye prochnosti oborudovaniya i truboprovodov atomnykh energeticheskikh ustanovok. Utochnennyy poverochnyy raschet na stadii proyektirovaniya [Substantiation of the strength of equipment and pipelines of nuclear power plants. An updated verification calculation at the design stage]. Moscow: Russian Institute of Standardization, 2021. 57 p.
ASME BPVC.III.5-2015. Section III. Rules for construction of nuclear facility components. Division 5. High Temperature Reactors. ASME, 2015. 500 p.
Gokhfel’d D.A. Nesushchaya sposobnost’ konstruktsiy v usloviyakh teplosmen. Moscow: Mashinostroyeniye, 1970. 260 p.
Gokhfel’d D.A., Chernyavskiy O.F. Nesushchaya sposobnost’ konstruktsiy pri povtornykh nagruzheniyakh. Moscow: Mashinostroyeniye, 1979. 263 p.
Chernyavsky O.F., Chernyavsky A.O. Limit states and safety factors under repeated loading. PNRPU Mechanics Bulletin. 2020. No. 3. P. 125–135. DOI: 10.15593/perm.mech/2020.3.12
Bree J. Elastic-plastic behaviour of thin tubes subjected to internal pressure and intermittent high-heat fluxes with application to fast-nuclear-reactor fuel elements. Journal of Strain Analysis. 1967. Vol. 2, no. 3. P. 226–238. DOI: 10.1243/03093247V023226
J. O.W., Porowski J. Upper bounds for accumulated strains due to creep ratcheting. Journal of Pressure Vessel Technology. 1974. Vol. 96, no. 3. P. 150–154.
Bradford R.A.W. The Bree problem with primary load cycling in-phase with the secondary load. International Journal of Pressure Vessels and Piping. 2012. No. 99. P. 44–50. DOI: 10.1016/j.ijpvp.2012.07.014
Bradford R.A.W. The Bree problem with primary load cycling out-of-phase with the secondary load. International Journal of Pressure Vessels and Piping. 2017. No. 154. P. 83–94. DOI: 10.1016/j.ijpvp.2017.06.004
Morozov N.F., Fedorenko R.V., Lukin A.V. Computational method for ratcheting analysis of the elastoplastic bodies under cyclic loads. Composites and Nanostructures. 2024. Vol. 16, no. 1. P. 62–78. DOI: 10.36236/1999-7590-2024-16-1-62-78
Pei X., Dong P., Mei J. The effects of kinematic hardening on thermal ratcheting and Bree diagram boundaries. Thin-Walled Structures. 2021. No. 159. 107235. DOI: 10.1016/j.tws.2020.107235
Fedorenko R.V., Lukin A.V., Murtazin I.R. The hardening type influence of pressure vessel ratcheting in case of thermal cyclic loads. PNRPU Mechanics Bulletin. 2025. No. 1. P. 117–128. DOI: 10.15593/perm.mech/2025.1.09
Bradford R.A.W., Ure J., Chen H.F. The Bree problem with different yield stresses on-load and off-load and application to creep ratcheting. International Journal of Pressure Vessels and Piping. 2014. No. 113. P. 32–39. DOI: 10.1016/j.ijpvp.2013.11.004
Bao H., Shen J., Liu Y., Chen H. Shakedown analysis and assessment method of four-stress parameters Bree-type problems. International Journal of Mechanical Sciences. 2022. No. 229. 107518. DOI: 10.1016/j.ijmecsci.2022.107518
Turkova V.A. Incremental analysis of twoaxial loading of the plate with central circular hole: Shakedown (accomodation), alternating plasticity, ratcheting. Vestnik of Samara State University. 2015. No. 3. P. 106–124.
Turkova V., Stepanova L. Finite element analysis of biaxial cyclic tensile loading of elasto-plastic plate with central elliptical hole. PNRPU Mechanics Bulletin. 2016. No. 3. P. 207–221. DOI: 10.15593/perm.mech/2016.3.14
Shi H., Chen G., Wang Y., Chen X. Ratcheting behavior of pressurized elbow pipe with local wall thinning. Int J. of Pressure Vessels and Piping. 2013. No. 102/103. P. 14–23. DOI: 10.12989/scs.2016.20.4.931
Zienkiewicz O.C. The finite element method in engineering science. London: McGraw-Hill, 1971. 521 p.
Maitournam M.H., Pommier B., Thomas J.-J. Determination of the asymptotic response of a structure under cyclic thermomechanical loading. Comptes Rendus. Mécanique. 2002. Vol. 330. P. 703–708. DOI: 10.1016/S1631-0721(02)01516-4
Abaqus Theory Guide (2016). URL: http://130.149.89.49:2080/v2016/books/stm/default.htm?startat=ch01s01ath01.html (accessed: 20.7.2025)
Krack M., Gross J. Harmonic Balance for Nonlinear Vibration Problems. 2019. 167 p. DOI: 10.1007/978-3-030-14023-6
Blahoš J., Vizzaccaro A., Salles L., El Haddad F. Parallel Harmonic Balance Method for Analysis of Nonlinear Dynamical Systems. Proceedings of the ASME Turbo Expo. 2020. P. 13. DOI: 10.1115/GT2020-15392
Martins T.S., Trainotti F., Zwölfer A., Zwölfer Z., Afonso F. A Python Implementation of a Robust Multi-Harmonic Balance With Numerical Continuation and Automatic Differentiation for Structural Dynamics. Journal of Computational and Nonlinear Dynamics. 2023. Vol. 18, no. 7. P. 12. DOI: 10.1115/1.4062424
Vizzaccaro A., Shen Y., Salles L., Blahoš J., Touzé C. Direct computation of nonlinear mapping via normal form for reduced-order models of finite element nonlinear structures. Computer Methods in Applied Mechanics and Engineering. 2020. Vol. 384. P. 36. DOI: 10.48550/arXiv.2009.12145
Raze G., Volvert M., Kerschen G. Tracking amplitude extrema of nonlinear frequency responses using the harmonic balance method. International Journal for Numerical Methods in Engineering. 2024. Vol. 125, no. 2. P. 28. DOI: 10.1002/nme.7376
Li Y.L., Huang J.L., Zhu W.D. A generalized incremental harmonic balance method by combining a data-driven framework for initial value selection of strongly nonlinear dynamic systems. International Journal of Non-Linear Mechanics. 2025. Vol. 169, no. 3/4. 104951. DOI: 10.1016/j.ijnonlinmec.2024.104951
Ladeveze P. Mécanique non linéaires des structures. Hermès, 1996. 280 p.
Bharali R. LATIN method for nonlinear dynamic problems. 2016. DOI: 10.13140/RG.2.2.28768.00005
Akel S., Nguyen Q.S. Determination of the limit response in cyclic plasticity. Computational Plasticity: Models, Software and Applications: Proceedings of the Second International Conference / ed. by O. D.R., H. E., O. E. 1989. P. 639–650.
Spiliopoulos K.V., Kapogiannis I.A. RSDM: A Powerful Direct Method to Predict the Asymptotic Cyclic Behavior of Elastoplastic Structures. Chinese Journal of Mechanical Engineering. 2021. Vol. 34, no. 140. P. 1–18. DOI: 10.1186/s10033-021-00658-0
Ishlinskiy A.Y.U. Obshchaya teoriya plastichnosti s lineynym uprochneniyem. Ukrainskiy matematicheskiy zhurnal. 1954. Vol. 6, no. 3. P. 314–324.
Kadashevich Y.I., Novozhilov V.V. The theory of plasticity with the Bauschinger effect taken into account. Doklady Akademii Nauk SSSR. 1957. Vol. 117, no. 4. P. 586–588.
Chaboche J.L. Time-independent constitutive theories for cyclic plasticity. International Journal of Plasticity. 1986. Vol. 2, no. 2. P. 149–188. URL: 10.1016/0749-6419(86)90010-0
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Computational Continuum Mechanics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
