A New Efficient Form of The Modified Energy Method (MEM) in Structural Dynamics

Document Type : Research


1 Ph.D. graduated of Structural Engineering, Department of Civil Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran.

2 Associate Professor, Department of Civil Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran.


The selection of a suitable numerical method to evaluate the dynamic behavior of structures, especially in nonlinear cases, is an important task in practice. Accordingly, the purpose of this study is to demonstrate the numerical features of a new single-step type of the Modified Energy Method (MEM) to compute the dynamic response of structural systems. A comprehensive formulation of this energy-based time integration scheme to incorporate the general nonlinear behavior in MDOF systems is presented for the first time ever in this paper. After discussing the stability and accuracy of the proposed time-stepping integration procedure, five applicable numerical examples in structural dynamics and earthquake engineering practices involving the various hysteretic behaviors and the effects of consistent mass and non-classical damping matrices are examined by the presented technique. In each case, the relevant comparisons are given in accordance to other available methods (e.g., Newmark and Runge-Kutta). Overall, the results indicate that the MEM yields a better accuracy than the 2nd Runge-Kutta approach. Furthermore, the distinguishing feature of the proposed method is to provide information about choosing the optimal size of the time intervals, especially in the nonlinear analyzes, which is not achievable in other applicable approaches.


1. Bathe, K.J., "Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme", Computers and Structures, vol. 85(7-8), 2007, p. 437-445. [DOI:10.1016/j.compstruc.2006.09.004]
2. Bathe, K.J., Cimento, A.P., "Some practical procedures for the solution of nonlinear finite element equations", Computer Methods in Applied Mechanics and Engineering, vol. 22(1), 1980, p. 59-85. [DOI:10.1016/0045-7825(80)90051-1]
3. Bathe, K.J., "Finite element procedures", Prentice Hall, Pearson Education, 2014.
4. Bathe, K.J., "On finite element methods for nonlinear dynamic response", In 7th European Conference on Structural Dynamics, 2008, p. 1239-1244.
5. Bayat, M., Pakar, I., Bayat, M., "High conservative nonlinear vibration equations by means of energy balance method", Earthquakes and Structures, vol. 11(1), 2016, p. 129-140. [DOI:10.12989/eas.2016.11.1.129]
6. Belytschko, T., Kam, W., Moran, B., Elkhodary, K.I., "Nonlinear Finite Elements for Continua and Structures", 2014.
7. Chang, S.Y., Tran, N.C., Wu, T.H., Yang, Y.S., "A One-Parameter Controlled Dissipative Unconditionally Stable Explicit Algorithm for Time History Analysis". Scientia Iranica, 2017. [DOI:10.24200/sci.2017.4158]
8. Chung, J., Hulbert, G.M., "A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method", Journal of Applied Mechanics, vol. 60(2), 1993, p. 371-375. [DOI:10.1115/1.2900803]
9. Cilsalar, H., Aydin, K., "Parabolic and cubic acceleration time integration schemes for nonlinear structural dynamics problems using the method of weighted residuals", Mechanics of Advanced Materials and Structures, vol. 23(7), 2016, p. 727-738. [DOI:10.1080/15376494.2015.1029162]
10. Clough, R., Penzien, J., "Dynamics of Structures", Berkeley, California, USA: Computers and Structures, 2013.
11. Farhat, C., Chapman, T., Avery, P., "Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models", International journal for numerical methods in engineering, (March), 2015, p. 1885-1891. [DOI:10.1002/nme.4820]
12. He, J., "Preliminary report on the energy balance for nonlinear oscillations", Mechanics Research Communications, vol.29(2-3), 2002, p. 107-111. [DOI:10.1016/S0093-6413(02)00237-9]
13. Hilber, H.M., Hughes, T.J.R., Taylor, R.L., "Improved numerical dissipation for time integration algorithms in structural dynamics", Earthquake Engineering & Structural Dynamics, vol. 5(3), 1997, p. 283-292. [DOI:10.1002/eqe.4290050306]
14. Houbolt, J.C., "A recurrence matrix solution for the dynamic response of elastic aircraft", Journal of the Aeronautical Sciences, vol. 17(9), 1950, p. 540-550. [DOI:10.2514/8.1722]
15. Humphreys, J.S., "On dynamic snap buckling of shallow arches", AIAA journal, vol. 4(5), 1966, p.878-886. [DOI:10.2514/3.3561]
16. Jalili Sadr Abad, M., Mahmoudi, M., Dowell, E.H., "Dynamic Analysis of SDOF Systems Using Modified Energy Method", Asian journal of civil engineering (BHRC), vol.18(7), 2017, p. 1125-1146.
17. Jalili Sadr Abad, M., Mahmoudi, M., Dowell, E.H., "Novel Technique for Dynamic Analysis of Shear-Frames Based on Energy Balance Equations", Scientia Iranica, 2018, p. 1-31. [DOI:10.24200/sci.2018.20790]
18. Khan, Y., Mirzabeigy, A., "Improved accuracy of He's energy balance method for analysis of conservative nonlinear oscillator", Neural Computing and Applications, vol. 25(3-4), 2014, p. 889-895. [DOI:10.1007/s00521-014-1576-2]
19. Kim, W., "Improved Time Integration Algorithms for the Analysis of Structural Dynamics", 2016.
20. Kim, W., Choi, S.Y., "An improved implicit time integration algorithm: The generalized composite time integration algorithm", Computers & Structures, vol. 196, 2018, p. 341-354. [DOI:10.1016/j.compstruc.2017.10.002]
21. Kolay, C., Ricles, J.M., "Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation", Earthquake Engineering & Structural Dynamics, vol. 43(9), 2014, p. 1361-1380. [DOI:10.1002/eqe.2401]
22. Kolay, C., Ricles, J.M., "Assessment of explicit and semi-explicit classes of model-based algorithms for direct integration in structural dynamics", International Journal for Numerical Methods in Engineering, vol. 107(1), 2016, p. 49-73. [DOI:10.1002/nme.5153]
23. Kuhl, D., Crisfield, M.A., "Energy-conserving and decaying algorithms in non-linear structural dynamics", International Journal for Numerical Methods in Engineering, vol. 45(November 1997), 1999, p. 569-599. https://doi.org/10.1002/(SICI)1097-0207(19990620)45:5<569::AID-NME595>3.0.CO;2-A [DOI:10.1002/(SICI)1097-0207(19990620)45:53.0.CO;2-A]
24. Mahmoudi, M., Montazeri, S., Jalili Sadr Abad, M., "Seismic performance of steel X-knee-braced frames equipped with shape memory alloy bars", Journal of Constructional Steel Research, vol. 147, 2018, p. 171-186. [DOI:10.1016/j.jcsr.2018.03.019]
25. Mazzoni, S., Mckenna, F., Scott, M.H., Fenves, G.L., "OpenSees Command Language Manual", 2007.
26. Navarro, H.A., Cveticanin, L., "Amplitude-frequency relationship obtained using Hamiltonian approach for oscillators with sum of non-integer order nonlinearities", Applied Mathematics and Computation, vol. 291, 2016, p. 162-171. [DOI:10.1016/j.amc.2016.06.047]
27. Newmark, N.M., "A method of computation for structural dynamics", Journal of the Engineering Mechanics Division, vol. 85(3), 1959, p. 67-94.
28. Park, K.C., "An improved stiffly stable method for direct integration of nonlinear structural dynamic equations", J. Appliled Mechanics, Trans. ASME, vol. 42(2), 1975, p. 464-470. [DOI:10.1115/1.3423600]
29. Razzak, M.A., Rahman, M.M., "Application of new novel energy balance method to strongly nonlinear oscillator systems", Results in Physics, vol. 5, 2015, p. 304-308. [DOI:10.1016/j.rinp.2015.10.001]
30. Reddy, J.N., "An Introduction to Nonlinear Finite Element Analysis: with applications to heat transfer, fluid mechanics, and solid mechanics", OUP Oxford, 2014. [DOI:10.1093/acprof:oso/9780199641758.001.0001]
31. Rostami, S., Shojaee, S., Moeinadini, A., "A parabolic acceleration time integration method for structural dynamics using quartic B-spline functions", Applied Mathematical Modelling, vol. 36(11), 2012, p. 5162-5182. [DOI:10.1016/j.apm.2011.11.047]
32. Shojaee, S., Rostami, S., Abbasi, A., "An unconditionally stable implicit time integration algorithm : Modified quartic B-spline method", computers and structures, vol. 153, 2015, p. 98-111. [DOI:10.1016/j.compstruc.2015.02.030]
33. Soroushian, A., "Integration Step Size and its Adequate Selection in Analysis of Structural Systems Against Earthquakes", In Computational Methods in Earthquake Engineering, Springer, 2017, p. 285-328. [DOI:10.1007/978-3-319-47798-5_10]
34. Weaver J.W., Johnston, P.R., "Structural dynamics by finite elements", Prentice-Hall Englewood Cliffs (NJ), 1987.
35. Wen, W.B., Wei, K., Lei, H.S., Duan, S.Y., Fang, D.N., "A novel sub-step composite implicit time integration scheme for structural dynamics", Computers & Structures, vol. 182, 2017, p. 176-186. [DOI:10.1016/j.compstruc.2016.11.018]
36. Wilson, E.L., Farhoomand, I., Bathe, K.J., Nonlinear dynamic analysis of complex structures. Earthquake Engineering & Structural Dynamics, 1(March 1972), 1973, p. 241-252. [DOI:10.1002/eqe.4290010305]
37. Zhang, L., Liu, T., Li, Q., "A Robust and Efficient Composite Time Integration Algorithm for Nonlinear Structural Dynamic Analysis", 2015. [DOI:10.1155/2015/907023]
38. Zienkiewicz, O.C., Taylor, R.L., Fox, D., "The Finite Element Method for Solid and Structural Mechanics", 2014.