Abstract
Recent advancements of performance-based wind design of tall buildings have placed increasing importance on effectively modeling of the nonlinear structural dynamic response under extreme storms. However, the numerical estimation of wind-induced nonlinear structural response based on the high-fidelity finite element model is computationally intensive due to its small time-step size and long simulation duration. To this end, the reduced-order modeling methodology using either physics-based analytical models or data-driven metamodels is widely applied to the simulation of nonlinear structural dynamics. With the rapid developments of machine learning techniques, the deep neural networks have recently become a popular data-driven approach for the efficient and accurate estimation of nonlinear structural responses. Due to a high demand on the quality and quantity of data, training the deep neural networks can become intractable. In this study, a knowledge-enhanced deep learning (KEDL) algorithm is proposed to simulate the wind-induced linear/nonlinear structural dynamic response. More specifically, the machine-readable knowledge in terms of both physics-based equations and/or semiempirical formulas is leveraged to enhance regularization mechanism during training of deep networks for structural dynamics. The KEDL methodology is data-efficient and robust to noise by effectively utilizing both the available input-output data and the prior knowledge on the structure of interest. In addition, the KEDL methodology is coupled with the wavelet-domain projection to simplify the input-output relationship, and hence to accelerate the training process. The data-efficient and noise-resistant characteristics of the KEDL methodology have been comprehensively investigated based on a single-degree-of-freedom (SDOF) system. Finally, it is clearly demonstrated that the trained knowledge-enhanced deep neural network presents both high simulation accuracy and computational efficiency in estimating the nonlinear dynamic response of a multidegree-of-freedom (MDOF) system under wind excitations.
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Data Availability Statement
Some or all data, models, or code generated or used during the study are available from the corresponding author by request (KEDL model).
Acknowledgments
The support for this project provided by NSF Grant No. CMMI 15-37431 is gratefully acknowledged.
References
Abadi, M., P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat, G. Irving, and M. Isard. 2016. “TensorFlow: A system for large-scale machine learning.” In Proc., 12th USENIX Symp. on Operating Systems Design and Implementation (OSDI 16), 265–283. Savannah, GA: The Advanced Computing Systems Association.
ASCE. 2016. Minimum design loads for buildings and other structures. Reston, VA: ASCE.
ASCE. 2019. Prestandard for performance-based wind design. Reston, VA: ASCE.
Bakir, T., B. Boussaid, P. F. Odgaard, M. N. Abdelkrim, and C. Aubrun. 2016. “Artificial neural network based on wavelet transform and feature extraction for a wind turbine diagnosis system.” In Proc., 14th Int. Conf. on Control, Automation, Robotics and Vision (ICARCV), 1–6. Piscataway, NJ: Institute of Electrical and Electronics Engineers.
Basu, B., and V. K. Gupta. 1998. “Seismic response of SDOF systems by wavelet modeling of nonstationary processes.” J. Eng. Mech. 124 (10): 1142–1150. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:10(1142).
Bengio, Y. 2009. “Learning deep architectures for AI.” Found. Trends Mach. Learn. 2 (1): 1–127. https://doi.org/10.1561/2200000006.
Bengio, Y., P. Simard, and P. Frasconi. 1994. “Learning long-term dependencies with gradient descent is difficult.” IEEE Trans. Neural Networks 5 (2): 157–166. https://doi.org/10.1109/72.279181.
Bu, H., D. Wang, P. Zhou, and H. Zhu. 2018. “An improved wavelet—Galerkin method for dynamic response reconstruction and parameter identification of shear-type frames.” J. Sound Vib. 419 (Apr): 140–157. https://doi.org/10.1016/j.jsv.2017.12.013.
Chaves, V., L. V. S. Sagrilo, and V. R. M. da Silva. 2015a. “Artificial neural networks applied to flexible pipes stochastic nonlinear dynamic analysis.” In Proc., 36th Iberian Latin American Congress on Computational Methods in Engineering, Rio de Janeiro, Brazil: Brazilian Association of Computational Methods in Engineering.
Chaves, V., L. V. S. Sagrilo, V. R. M. da Silva, and M. A. Vignoles. 2015b. “Artificial neural networks applied to flexible pipes fatigue calculations.” In Proc., 34th Int. Conf. on Ocean, Offshore and Arctic Engineering. New York: ASME.
Chen, J., and J. Li. 2004. “Simultaneous identification of structural parameters and input time history from output-only measurements.” Comput. Mech. 33 (5): 365–374. https://doi.org/10.1007/s00466-003-0538-9.
Chen, X. 2013. “Estimation of stochastic crosswind response of wind-excited tall buildings with nonlinear aerodynamic damping.” Eng. Struct. 56 (Nov): 766–778. https://doi.org/10.1016/j.engstruct.2013.05.044.
Chen, X. 2014. “Extreme value distribution and peak factor of crosswind response of flexible structures with nonlinear aeroelastic effect.” J. Struct. Eng. 140 (12): 04014091. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001017.
Chuang, W. C., and S. M. J. Spence. 2017. “A performance-based design framework for the integrated collapse and non-collapse assessment of wind excited buildings.” Eng. Struct. 150 (Nov): 746–758. https://doi.org/10.1016/j.engstruct.2017.07.030.
Cui, W., and L. Caracoglia. 2018a. “A unified framework for performance-based wind engineering of tall buildings in hurricane-prone regions based on lifetime intervention-cost estimation.” Struct. Saf. 73 (Jul): 75–86. https://doi.org/10.1016/j.strusafe.2018.02.003.
Cui, W., and L. Caracoglia. 2018b. “A fully-coupled generalized model for multi-directional wind loads on tall buildings: A development of the quasi-steady theory.” J. Fluids Struct. 78 (Apr): 52–68. https://doi.org/10.1016/j.jfluidstructs.2017.12.008.
Cybenko, G. 1989. “Approximation by superpositions of a sigmoidal function.” Math. Cont. Signals Syst. 2 (4): 303–314. https://doi.org/10.1007/BF02551274.
Daubechies, I. 1993. “Orthogonal bases of compactly supported wavelets. II: Variations on a theme.” SIAM J. Math. Anal. 24 (2): 499–519. https://doi.org/10.1137/0524031.
Deodatis, G. 1996. “Simulation of ergodic multivariate stochastic processes.” J. Eng. Mech. 122 (8): 778–787. https://doi.org/10.1061/(ASCE)0733-9399(1996)122:8(778).
Derkevorkian, A., M. Hernandez-Garcia, H. Yun, S. F. Masri, and P. Li. 2015. “Nonlinear data-driven computational models for response prediction and change detection.” Struct. Control Health Monit. 22 (2): 273–288. https://doi.org/10.1002/stc.1673.
Di Matteo, A., F. Lo Iacono, G. Navarra, and A. Pirrotta. 2014. “Experimental validation of a direct pre-design formula for TLCD.” Eng. Struct. 75 (Apr): 528–538. https://doi.org/10.1016/j.engstruct.2014.05.045.
Dissanayake, M. W. M. G., and N. Phan-Thien. 1994. “Neural-network-based approximations for solving partial differential equations.” Commun. Numer. Methods Eng. 10 (3): 195–201. https://doi.org/10.1002/cnm.1640100303.
Duda, R. O., P. E. Hart, and D. G. Stork. 2012. Pattern classification. New York: Wiley.
Elfadel, I. A. M., D. S. Boning, and X. Li. 2018. Machine learning in VLSI computer-aided design. Berlin: Springer.
Facchini, L., M. Betti, and P. Biagini. 2014. “Neural network based modal identification of structural systems through output-only measurement.” Comput. Struct. 138 (1): 183–194. https://doi.org/10.1016/j.compstruc.2014.01.013.
Feng, C., and X. Chen. 2017. “Crosswind response of tall buildings with nonlinear aerodynamic damping and hysteretic restoring force character.” J. Wind Eng. Ind. Aerodyn. 167 (8): 62–74. https://doi.org/10.1016/j.jweia.2017.04.012.
Feng, C., and X. Chen. 2018. “Inelastic responses of wind-excited tall buildings: Improved estimation and understanding by statistical linearization approaches.” Eng. Struct. 159 (Apr): 141–154. https://doi.org/10.1016/j.engstruct.2017.12.041.
Gers, F. A., J. Schmidhuber, and F. Cummins. 1999. “Learning to forget: Continual prediction with LSTM.” In 9th Int. Conf. on Artificial Neural Networks, 1–19. London: IET.
Ghanem, R., and F. Romeo. 2000. “A wavelet-based approach for the identification of linear time-varying dynamical systems.” J. Sound Vib. 234 (4): 555–576. https://doi.org/10.1006/jsvi.1999.2752.
Ghanem, R., and F. Romeo. 2001. “A wavelet-based approach for model and parameter identification of non-linear systems.” Int. J. Non Linear Mech. 36 (5): 835–859. https://doi.org/10.1016/S0020-7462(00)00050-0.
Gholizadeh, S., J. Salajegheh, and E. Salajegheh. 2009. “An intelligent neural system for predicting structural response subject to earthquakes.” Adv. Eng. Software 40 (8): 630–639. https://doi.org/10.1016/j.advengsoft.2008.11.008.
Goodfellow, I., Y. Bengio, and A. Courville. 2016. Deep learning. Cambridge, MA: MIT Press.
Graves, A., A. R. Mohamed, and G. Hinton. 2013. “Speech recognition with deep recurrent neural networks.” In Proc., 2013 IEEE Int. Conf. on Acoustics, Speech and Signal Processing, 6645–6649. Piscataway, NJ: IEEE.
Guarize, R., N. A. F. Matos, L. V. S. Sagrilo, and E. C. P. Lima. 2007. “Neural networks in the dynamic response analysis of slender marine structures.” Appl. Ocean Res. 29 (4): 191–198. https://doi.org/10.1016/j.apor.2008.01.002.
He, K., X. Zhang, S. Ren, and J. Sun. 2015. “Delving deep into rectifiers: Surpassing human-level performance on ImageNet classification.” Preprint, submitted February 6, 2015. http://arxiv.org/abs/1502.01852.
Hinton, G. E., S. Osindero, and Y. W. Teh. 2006. “A fast learning algorithm for deep belief nets.” Neural Comput. 18 (7): 1527–1554. https://doi.org/10.1162/neco.2006.18.7.1527.
Hochreiter, S., and J. Schmidhuber. 1997. “Long short-term memory.” Neural Comput. 9 (8): 1735–1780. https://doi.org/10.1162/neco.1997.9.8.1735.
Holmes, J. D. 2015. Wind loading of structures, 3rd ed. Boca Raton, FL: CRC Press.
Huang, G., X. Chen, H. Liao, and M. Li. 2013. “Predicting of tall building response to nonstationary winds using multiple wind speed samples.” Wind Struct. Int. J. 17 (2): 227–244. https://doi.org/10.12989/was.2013.17.2.227.
Iungo, G. V., C. Santoni-Ortiz, M. Abkar, F. Porté-Agel, M. A. Rotea, and S. Leonardi. 2015. “Data-driven reduced order model for prediction of wind turbine wakes.” J. Phys. Conf. Ser. 625 (1): 012009.
Jia, F., Y. Lei, J. Lin, X. Zhou, and N. Lu. 2016. “Deep neural networks: A promising tool for fault characteristic mining and intelligent diagnosis of rotating machinery with massive data.” Mech. Syst. Sig. Process. 72 (May): 303–315. https://doi.org/10.1016/j.ymssp.2015.10.025.
Jiang, X., and H. Adeli. 2005. “Dynamic wavelet neural network for nonlinear identification of high rise buildings.” Comput.-Aided Civ. Infrastruct. Eng. 20 (5): 316–330. https://doi.org/10.1111/j.1467-8667.2005.00399.x.
Joyner, M. D., and M. Sasani. 2018. “Multihazard risk-based resilience analysis of east and west coast buildings designed to current codes.” J. Struct. Eng. 144 (9): 04018156. https://doi.org/10.1061/(ASCE)ST.1943-541X.0002132.
Jozefowicz, R., W. Zaremba, and I. Sutskever. 2015. “An empirical exploration of recurrent network architectures.” In Proc., of 32nd Int. Conference on Machine Learning 2342–2350. Lille, France: International Conference on Machine Learning.
Kareem, A., and T. Wu. 2013. “Wind-induced effects on bluff bodies in turbulent flows: Nonstationary, non-Gaussian and nonlinear features.” J. Wind Eng. Ind. Aerodyn. 122 (Nov): 21–37. https://doi.org/10.1016/j.jweia.2013.06.002.
Khallaf, M., and J. Jupp. 2017. “Performance-based design of tall building envelopes using competing wind load and wind flow criteria.” In Proc., Int. High-Performance Built Environment Conf., 99–109. Amsterdam, Netherlands: Elsevier.
Kingma, D. P., and J. Ba. 2014. “Adam: A method for stochastic optimization.” Preprint, submitted December 22, 2014. http://arxiv.org/abs/1412.6980.
Kwon, D. K., and A. Kareem. 2013. “Generalized gust-front factor: A computational framework for wind load effects.” Eng. Struct. 48 (Mar): 635–644. https://doi.org/10.1016/j.engstruct.2012.12.024.
Le, T. H., and L. Caracoglia. 2015. “Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds.” J. Sound Vib. 344 (May): 179–208. https://doi.org/10.1016/j.jsv.2015.01.007.
LeCun, Y., Y. Bengio, and G. Hinton. 2015. “Deep learning.” Nature 521 (7553): 436–444. https://doi.org/10.1038/nature14539.
Li, T., T. Wu, and Z. Liu. 2020. “Nonlinear unsteady bridge aerodynamics: Reduced-order modeling based on deep LSTM networks.” J. Wind Eng. Ind. Aerodyn. 198 (Mar): 104116. https://doi.org/10.1016/j.jweia.2020.104116.
Linse, D. J., and R. F. Stengel. 1993. “Identification of aerodynamic coefficients using computational neural networks.” J. Guidance Control Dyn. 16 (6): 1018–1025. https://doi.org/10.2514/3.21122.
Lu, X., X. Lu, H. Guan, and L. Ye. 2013. “Comparison and selection of ground motion intensity measures for seismic design of super high-rise buildings.” Adv. Struct. Eng. 16 (7): 1249–1262. https://doi.org/10.1260/1369-4332.16.7.1249.
Luo, J., N. E. Wierschem, L. A. Fahnestock, B. F. Spencer, D. D. Quinn, D. M. McFarland, A. F. Vakakis, and L. A. Bergman. 2014. “Design, simulation, and large-scale testing of an innovative vibration mitigation device employing essentially nonlinear elastomeric springs.” Earthquake Eng. Struct. Dyn. 43 (12): 1829–1851. https://doi.org/10.1002/eqe.2424.
Mao, Z., and M. Todd. 2012. “A model for quantifying uncertainty in the estimation of noise-contaminated measurements of transmissibility.” Mech. Syst. Sig. Process. 28 (Apr): 470–481. https://doi.org/10.1016/j.ymssp.2011.10.002.
McCulloch, W. S., and W. Pitts. 1943. “A logical calculus of the ideas immanent in nervous activity.” Bull. Math. Biophys. 5 (4): 115–133. https://doi.org/10.1007/BF02478259.
Mead, C., and M. Ismail. 2012. Analog VLSI implementation of neural systems. Berlin: Springer.
Pascanu, R., C. Gulcehre, K. Cho, and Y. Bengio. 2013. “How to construct deep recurrent neural networks.” Preprint, submitted December 20, 2013. http://arxiv.org/abs/1312.6026.
Peherstorfer, B., and K. Willcox. 2015. “Dynamic data-driven reduced-order models.” Comput. Methods Appl. Mech. Eng. 291 (Jul): 21–41. https://doi.org/10.1016/j.cma.2015.03.018.
Pei, J. S., J. P. Wright, and A. W. Smyth. 2005. “Mapping polynomial fitting into feedforward neural networks for modeling nonlinear dynamic systems and beyond.” Comput. Methods Appl. Mech. Eng. 194 (42): 4481–4505. https://doi.org/10.1016/j.cma.2004.12.010.
Petersen, P., and F. Voigtlaender. 2018. “Optimal approximation of piecewise smooth functions using deep ReLU neural networks.” Neural Networks 108 (Dec): 296–330. https://doi.org/10.1016/j.neunet.2018.08.019.
Psichogios, D. C., and L. H. Ungar. 1992. “A hybrid neural network-first principles approach to process modeling.” AlChE J. 38 (10): 1499–1511. https://doi.org/10.1002/aic.690381003.
Raissi, M., P. Perdikaris, and G. E. Karniadakis. 2017a. “Numerical Gaussian processes for time-dependent and non-linear partial differential equations.” Preprint, submitted March 29, 2017. http://arxiv.org/abs/1703.10230.
Raissi, M., P. Perdikaris, and G. E. Karniadakis. 2017b. “Physics informed deep learning. Part I: Data-driven solutions of nonlinear partial differential equations.” Preprint, submitted November 28, 2017. http://arxiv.org/abs/1711.10561.
Romine, C., and B. Peyton. 1997. Computing connection coefficients of compactly supported wavelets on bounded intervals. Oak Ridge, TN: Oak Ridge National Laboratory.
Rosenblatt, F. 1958. “The perceptron: A probabilistic model for information storage and organization in the brain.” Psychol. Rev. 65 (6): 386–408. https://doi.org/10.1037/h0042519.
Rumelhart, D. E., G. E. Hinton, and R. J. Williams. 1985. Learning internal representations by error propagation. La Jolla, CA: California Univ., San Diego, La Jolla. Inst for Cognitive Science.
Saitua, F., D. Lopez-Garcia, and A. A. Taflanidis. 2018. “Optimization of height-wise damper distributions considering practical design issues.” Eng. Struct. 173 (Oct): 768–786. https://doi.org/10.1016/j.engstruct.2018.04.008.
Saravanan, N., and K. I. Ramachandran. 2010. “Incipient gear box fault diagnosis using discrete wavelet transform (DWT) for feature extraction and classification using artificial neural network (ANN).” Expert Syst. Appl. 37 (6): 4168–4181. https://doi.org/10.1016/j.eswa.2009.11.006.
Sezer, O. B., and A. M. Ozbayoglu. 2018. “Algorithmic financial trading with deep convolutional neural networks: Time series to image conversion approach.” Appl. Soft Comput. 70 (Sep): 525–538. https://doi.org/10.1016/j.asoc.2018.04.024.
Siringoringo, D. M., and Y. Fujino. 2017. “Wind-induced responses and dynamics characteristics of an asymmetrical base-isolated building observed during typhoons.” J. Wind Eng. Ind. Aerodyn. 167 (Aug): 183–197. https://doi.org/10.1016/j.jweia.2017.04.020.
Snaiki, R., and T. Wu. 2019. “Knowledge-enhanced deep learning for simulation of tropical cyclone boundary-layer winds.” J. Wind Eng. Ind. Aerodyn. 194 (Nov): 103983. https://doi.org/10.1016/j.jweia.2019.103983.
Spence, S. M. J., and A. Kareem. 2014. “Performance-based design and optimization of uncertain wind-excited dynamic building systems.” Eng. Struct. 78 (Nov): 133–144. https://doi.org/10.1016/j.engstruct.2014.07.026.
Spiridonakos, M. D., and E. N. Chatzi. 2015. “Metamodeling of dynamic nonlinear structural systems through polynomial chaos NARX models.” Comput. Struct. 157 (Sep): 99–113. https://doi.org/10.1016/j.compstruc.2015.05.002.
Subasi, A. 2007. “EEG signal classification using wavelet feature extraction and a mixture of expert model.” Expert Syst. Appl. 32 (4): 1084–1093. https://doi.org/10.1016/j.eswa.2006.02.005.
Suksuwan, A., and S. M. J. Spence. 2018. “Performance-based multi-hazard topology optimization of wind and seismically excited structural systems.” Eng. Struct. 172 (Oct): 573–588. https://doi.org/10.1016/j.engstruct.2018.06.039.
Sun, H., and O. Buyukozturk. 2018. “The MIT green building benchmark problem for structural health monitoring of tall buildings.” Struct. Control Health Monit. 25 (3): 17. https://doi.org/10.1002/stc.2115.
Sun, Z., and C. Chang. 2004. “Statistical wavelet-based method for structural health monitoring.” J. Struct. Eng. 130 (7): 1055–1062. https://doi.org/10.1061/(ASCE)0733-9445(2004)130:7(1055).
Tabbuso, P., S. M. J. Spence, L. Palizzolo, A. Pirrotta, and A. Kareem. 2016. “An efficient framework for the elasto-plastic reliability assessment of uncertain wind excited systems.” Struct. Saf. 58 (Jan): 69–78. https://doi.org/10.1016/j.strusafe.2015.09.001.
Tamura, Y., and A. Kareem. 2013. Advanced structural wind engineering. Berlin: Springer.
Tamura, Y., R. Kohsaka, O. Nakamura, K. I. Miyashita, and V. J. Modi. 1996. “Wind-induced responses of an airport tower—Efficiency of tuned liquid damper.” J. Wind Eng. Ind. Aerodyn. 65 (1): 121–131. https://doi.org/10.1016/S0167-6105(97)00028-7.
Vaidyanathan, C. V., P. Kamatchi, and R. Ravichandran. 2005. “Artificial neural networks for predicting the response of structural systems with viscoelastic dampers.” Comput. Aided Civ. Infrastruct. Eng. 20 (4): 294–302. https://doi.org/10.1111/j.1467-8667.2005.00395.
Walt, S. V. D., S. C. Colbert, and G. Varoquaux. 2011. “The NumPy array: A structure for efficient numerical computation.” Comput. Sci. Eng. 13 (2): 22–30. https://doi.org/10.1109/MCSE.2011.37.
Wang, S., and S. A. Mahin. 2018. “High-performance computer-aided optimization of viscous dampers for improving the seismic performance of a tall building.” Soil Dyn. Earthquake Eng. 113 (Oct): 454–461. https://doi.org/10.1016/j.soildyn.2018.06.008.
Wu, R. T., and M. R. Jahanshahi. 2019. “Deep convolutional neural network for structural dynamic response estimation and system identification.” J. Eng. Mech. 145 (1): 04018125. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001556.
Wu, T., and A. Kareem. 2011. “Modeling hysteretic nonlinear behavior of bridge aerodynamics via cellular automata nested neural network.” J. Wind Eng. Ind. Aerodyn. 99 (4): 378–388. https://doi.org/10.1016/j.jweia.2010.12.011.
Wu, T., A. Kareem, and Y. Ge. 2013. “Linear and nonlinear aeroelastic analysis frameworks for cable-supported bridges.” Nonlinear Dyn. 74 (3): 487–516. https://doi.org/10.1007/s11071-013-0984-7.
Xiong, C., X. Lu, H. Guan, and Z. Xu. 2016. “A nonlinear computational model for regional seismic simulation of tall buildings.” Bull. Earthquake Eng. 14 (4): 1047–1069. https://doi.org/10.1007/s10518-016-9880-0.
Yang Y., M. Hou, H. Sun, T. Zhang, F. Weng, and J. Luo. 2019. “Neural network algorithm based on Legendre improved extreme learning machine for solving elliptic partial differential equations.” Soft Comput. 24 (2): 1–14. https://doi.org/10.1007/s00500-019-03944-1.
Zeng, X., Y. Peng, and J. Chen. 2017. “Serviceability-based damping optimization of randomly wind-excited high-rise buildings.” Struct. Des. Tall Spec. Build. 26 (11): e1371. https://doi.org/10.1002/tal.1371.
Zhang, L., X. Lu, H. Guan, L. Xie, and X. Lu. 2017. “Floor acceleration control of super tall buildings with vibration reduction substructures.” Struct. Des. Tall Spec. Build. 26 (16): 13. https://doi.org/10.1002/tal.1343.
Zhang, W., and Y. Xu. 2001. “Closed form solution for along-wind response of actively controlled tall buildings with LQG controllers.” J. Wind Eng. Ind. Aerodyn. 89 (9): 785–807. https://doi.org/10.1016/S0167-6105(01)00068-X.
Zhang, Y., L. Li, Y. Guo, and X. Zhang. 2018. “Bidirectional wind response control of 76-story benchmark building using active mass damper with a rotating actuator.” Struct. Control Health Monit. 25 (10): e2216. https://doi.org/10.1002/stc.2216.
Zhao, N., G. Huang, Q. Yang, X. Zhou, and A. Kareem. 2019. “Fast convolution integration–based nonstationary response analysis of linear and nonlinear structures with nonproportional damping.” J. Eng. Mech. 145 (8): 04019053. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001633.
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Journal of Structural Engineering
Volume 146 • Issue 11 • November 2020
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© 2020 American Society of Civil Engineers.
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Received: Oct 7, 2019
Accepted: May 27, 2020
Published online: Aug 21, 2020
Published in print: Nov 1, 2020
Discussion open until: Jan 21, 2021
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