Exact Sensitivity of Nonlinear Dynamic Response with Modal and Rayleigh Damping Formulated with the Tangent Stiffness
Publication: Journal of Structural Engineering
Volume 150, Issue 3
Abstract
Derivatives of nonlinear dynamic response are calculated in an exact and efficient manner with respect to material, geometry, mass, and damping parameters. Developments are presented for the Rayleigh and modal damping options that use the updated tangent stiffness. The calculation of exact response sensitivities for those options requires the calculation of derivatives of eigenvalues and eigenvectors; it is also shown that the third-order tensor formed by the derivative of the stiffness matrix with respect to the displacement vector is needed. That tensor, which amends the coefficient matrix of the system of equations that governs the response sensitivities, is nonzero for materials with continuously varying stiffness. The Bouc–Wen material model exhibits that feature and is selected to demonstrate the developments. Correct differentiation and assembly at the material, section, and element levels are highlighted. The results, verified by finite difference, suggest that the sensitivity of the response is influenced by the choice of damping model, strongly for some parameters, particularly when higher modes contribute to the nonlinear structural behavior.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or code generated or used during the study are available from the corresponding author by request. (All Python code, including examples.)
References
Arora, J., and J. Cardoso. 1989. “A design sensitivity analysis principle and its implementation into ADIANA.” Comput. Struct. 32 (3–4): 691–705. https://doi.org/10.1016/0045-7949(89)90357-X.
Baber, T. T., and M. N. Noori. 1985. “Random vibration of degrading, pinching systems.” J. Eng. Mech. 111 (8): 1010–1026. https://doi.org/10.1061/(ASCE)0733-9399(1985)111:8(1010).
Baber, T. T., and Y.-K. Wen. 1981. “Random vibration of hysteretic, degrading systems.” J. Eng. Mech. Div. 107 (6): 1069–1087. https://doi.org/10.1061/JMCEA3.0002768.
Cardoso, J., and J. Arora. 1992. “Design sensitivity analysis of nonlinear dynamic response of structural and mechanical systems.” Struct. Optim. 4 (Mar): 37–46. https://doi.org/10.1007/BF01894079.
Casciati, F. 1989. “Stochastic dynamics of hysteretic media.” Struct. Saf. 6 (2–4): 259–269. https://doi.org/10.1016/0167-4730(89)90026-X.
Charney, F. A. 2008. “Unintended consequences of modeling damping in structures.” J. Struct. Eng. 134 (4): 581–592. https://doi.org/10.1061/(ASCE)0733-9445(2008)134:4(581).
Choi, K. K., J. L. Santos, and T.-M. Yao. 1988. “Recent advances in design sensitivity analysis and its use in structural design process.” SAE Trans. 97 (8): 181–197.
Chopra, A. K., and F. McKenna. 2016. “Modeling viscous damping in nonlinear response history analysis of buildings for earthquake excitation.” Earthquake Eng. Struct. Dyn. 45 (2): 193–211. https://doi.org/10.1002/eqe.2622.
Conte, J. 2017. “Finite element response sensitivity analysis in earthquake engineering.” In Earthquake engineering frontiers in the new millennium, edited by B. F. Spencer Jr and Y. X. Hu, 395–401. Tokyo: A.A. Balkema.
Conte, J. P., P. Vijalapura, and M. Meghella. 2003. “Consistent finite-element response sensitivity analysis.” J. Eng. Mech. 129 (12): 1380–1393. https://doi.org/10.1061/(ASCE)0733-9399(2003)129:12(1380).
Fox, R., and M. Kapoor. 1968. “Rates of change of eigenvalues and eigenvectors.” AIAA J. 6 (12): 2426–2429. https://doi.org/10.2514/3.5008.
Gao, Y., Q. Gu, Z. Qiu, and J. Wang. 2016. “Seismic response sensitivity analysis of coupled dam-reservoir-foundation systems.” J. Eng. Mech. 142 (10): 04016070. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001124.
Gu, Q., J. P. Conte, A. Elgamal, and Z. Yang. 2009. “Finite element response sensitivity analysis of multi-yield-surface plasticity model by direct differentiation method.” Comput. Methods Appl. Mech. Eng. 198 (30–32): 2272–2285. https://doi.org/10.1016/j.cma.2009.02.030.
Hall, J. F. 2006. “Problems encountered from the use (or misuse) of Rayleigh damping.” Earthquake Eng. Struct. Dyn. 35 (5): 525–545. https://doi.org/10.1002/eqe.541.
Haukaas, T. 2023. “Sensitivity to damping in nonlinear dynamics.” In Proc., 9th ECCOMAS Thematic Conf. on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN). Atlanta: International Association of Computational Mechanics.
Haukaas, T., and A. Der Kiureghian. 2005. “Parameter sensitivity and importance measures in nonlinear finite element reliability analysis.” J. Eng. Mech. 131 (10): 1013–1026. https://doi.org/10.1061/(ASCE)0733-9399(2005)131:10(1013).
Kleiber, M., T. D. Hien, H. Antúnez, and P. Kowalczyk. 1997. Parameter sensitivity in nonlinear mechanics: Theory and finite element computations. West Sussex, UK: Wiley.
Magnus, J. R. 1985. “On differentiating eigenvalues and eigenvectors.” Econ. Theory 1 (2): 179–191. https://doi.org/10.1017/S0266466600011129.
McKenna, F., M. H. Scott, and G. L. Fenves. 2010. “Nonlinear finite-element analysis software architecture using object composition.” J. Comput. Civ. Eng. 24 (1): 95–107. https://doi.org/10.1061/(ASCE)CP.1943-5487.0000002.
Nelson, R. B. 1976. “Simplified calculation of eigenvector derivatives.” AIAA J. 14 (9): 1201–1205. https://doi.org/10.2514/3.7211.
Newmark, N. M. 1959. “A method of computation for structural dynamics.” J. Eng. Mech. Div. 85 (3): 67–94. https://doi.org/10.1061/JMCEA3.0000098.
Puthanpurayil, A. M., A. J. Carr, and R. P. Dhakal. 2018. “Application of nonlocal elasticity continuum damping models in nonlinear dynamic analysis.” Bull. Earthquake Eng. 16 (Jun): 6269–6297. https://doi.org/10.1007/s10518-018-0412-y.
Puthanpurayil, A. M., O. Lavan, A. J. Carr, and R. P. Dhakal. 2016. “Elemental damping formulation: An alternative modelling of inherent damping in nonlinear dynamic analysis.” Bull. Earthquake Eng. 14 (Mar): 2405–2434. https://doi.org/10.1007/s10518-016-9904-9.
Schellenberg, A., A. Sarebanha, M. Schoettler, G. Mosqueda, G. Benzoni, and S. Mahin. 2015. Hybrid simulation of seismic isolation systems applied to an apr-1400 nuclear power plant. Berkeley, CA: University of California, Berkeley.
Scott, M. H. 2019. “Be careful with modal damping.” Portwood Digital Blog. Accessed March 16, 2023. https://portwooddigital.com/2019/09/12/be-careful-with-modal-damping.
Scott, M. H., P. Franchin, G. L. Fenves, and F. C. Filippou. 2004. “Response sensitivity for nonlinear beam–column elements.” J. Struct. Eng. 130 (9): 1281–1288. https://doi.org/10.1061/(ASCE)0733-9445(2004)130:9(1281).
Tsay, J., J. Cardoso, and J. Arora. 1990a. “Nonlinear structural design sensitivity analysis for path dependent problems. Part 1: General theory.” Comput. Methods Appl. Mech. Eng. 81 (2): 183–208. https://doi.org/10.1016/0045-7825(90)90109-Y.
Tsay, J., J. Cardoso, and J. Arora. 1990b. “Nonlinear structural design sensitivity analysis for path dependent problems. Part 2: Analytical examples.” Comput. Methods Appl. Mech. Eng. 81 (2): 209–228. https://doi.org/10.1016/0045-7825(90)90110-8.
Wen, Y.-K. 1976. “Method for random vibration of hysteretic systems.” J. Eng. Mech. Div. 102 (2): 249–263. https://doi.org/10.1061/JMCEA3.0002106.
Zhang, Y., and A. Der Kiureghian. 1993. “Dynamic response sensitivity of inelastic structures.” Comput. Methods Appl. Mech. Eng. 108 (1–2): 23–36. https://doi.org/10.1016/0045-7825(93)90151-M.
Information & Authors
Information
Published In
Journal of Structural Engineering
Volume 150 • Issue 3 • March 2024
Copyright
© 2024 American Society of Civil Engineers.
History
Received: Mar 23, 2023
Accepted: Oct 30, 2023
Published online: Jan 16, 2024
Published in print: Mar 1, 2024
Discussion open until: Jun 16, 2024
ASCE Technical Topics:
- Continuum mechanics
- Damping
- Dynamic response
- Dynamics (solid mechanics)
- Engineering fundamentals
- Engineering mechanics
- Material mechanics
- Material properties
- Materials engineering
- Mathematical functions
- Mathematics
- Matrix (mathematics)
- Nonlinear response
- Parameters (statistics)
- Solid mechanics
- Statistics
- Stiffening
- Structural behavior
- Structural dynamics
- Structural engineering
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.