Novel technique for dynamic analysis of shear frames based on energy balance equations

Document Type : Article

Authors

1 Department of Civil Engineering, Shahid Rajaee Teacher Training University, Tehran, P.O. Box 16788-15844, Iran

2 Department of Mechanical Engineering and Materials Science, Duke University, Durham, USA

Abstract

In this paper, an efficient computational solution technique based on the energy balance equations is presented for the dynamic analysis of shear-frames, as an example of a multi-degree-of-freedom system. After deriving the dynamic energy balance equations for these systems, a new mathematical solution technique which is called Elimination of Discontinuous Velocities is proposed to solve a set of coupled quadratic algebraic equations. The method will be illustrated for the free vibration of a two-story structure. Subsequently, the damped dynamic response of a three-story shear-frame which is subjected to harmonic loading is considered. Finally the analysis of a three-story shear-building under horizontal earthquake load, as one of the most common problems in Earthquake Engineering, is studied. The results show that this method has acceptable accuracy in comparison with other techniques, but is not only faster compared with modal analysis but also does not require adjusting and calibrating the stability parameter as compared with a method of time integration like the Newmark method.

Keywords

Main Subjects


1. Wilson, E.L. Dynamic analysis", In Three- Dimensional Static and Dynamic Analysis of Structures, 3rd Edn., pp. 165{175, Computers and Structures, Berkeley, California, USA (2002). 2. Craig, R. and Kurila, A. Preface to structural dynamics", In Fundamentals of Structural Dynamics, 2nd Edn., pp. 13{14, Wiley, New Jersey, USA (2006). 3. Clough, R. and Penzien, J. Earthquake engineering", In Dynamics of Structures, 3rd Edn., pp. 555{730, Computers and Structures, Berkeley, California, USA (2013). 4. Chopra, A.K. Earthquake analysis of arch dams: factors to be considered", ASCE, 138(2), pp. 205{214 (2012). 5. Chopra, A.K. Earthquake analysis of concrete dams: factors to be considered", 10th U.S. National Conference on Earthquake Engineering, Anchorage, Alaska, USA (2014). 6. Feng, M.Q., Fukuda, Y., Feng, D., and Mizuta, M. Nontarget vision sensor for remote measurement of bridge dynamic response", J. Bridg. Eng., 20(2015), pp. 1{12 (2015). 1106 M. Jalili Sadr Abad et al./Scientia Iranica, Transactions A: Civil Engineering 27 (2020) 1091{1112 7. Krodkiewski, J.M., Mechanical Vibration, Melbourne University, pp. 1{247, Melbourne, Australia (2008). 8. Chopra, A.K. Single degree-of-freedom systems", In Dynamics of Structures, 4th Edn., pp. 1{307, Prentice Hall, Berkeley, University of California, USA (2013). 9. Makem, J.E., Armstrong, C.G., and Robinson, T.T. Automatic decomposition and e_cient semistructured meshing of complex solids", Engineering with Computers, 30(3), pp. 345{361 (2014). 10. Kougioumtzoglou, I.A. and Spanos P.D. Nonlinear MDOF system stochastic response determination via a dimension reduction approach", Comput. Struct., 126(1), pp. 135{148 (2013). 11. Brahmi, K., Bouhaddi, N., and Fillod, R., Reduction of Junction Degree of Freedom in Certain Methods of Dynamic Substructure Synthesis, The International Society for Optical Engineering, Spie International Society for Optical, Bellingham, USA, pp. 1763{1763 (1995). 12. Kreyszig, E. Numerical analysis", In Advanced Engineering Mathematics, 10th Edn., Wiley, pp. 787{949, New York, USA (2005). 13. Blahut, E.R. Fast algorithms for the discrete Fourier transform", In Fast Algorithms for Signal Processing, pp. 68{114, Cambridge University Press, New York, USA (2010). 14. Elizalde-Siller, H.R. Non-linear modal analysis methods for engineering structures", Doctoral dissertation, Imperial College London, UK (2004). 15. Wong, K.K.F. Nonlinear dynamic analysis of structures using modal superposition", ASCE Structures Congress 2011, pp. 770{781, Las Vegas, Nevada, USA (2011). 16. Dou, S. and Jakob, S.J. Optimization of hardening/ softening behavior of plane frame structures using nonlinear normal modes", Comput. Struct., 164, pp. 63{74 (2016). 17. Dou, S. Gradient-based optimization in nonlinear structural dynamics", PhD Thesis, Mechanical Engineering, DTU, Denmark (2015). 18. Dou, S. and Jakob, S.J. Analytical sensitivity analysis and topology optimization of nonlinear resonant structures with hardening and softening behavior", 17th U.S. National Congress on Theoretical and Applied Mechanics, East Lansing, Michigan, USA, pp. 1{3 (2014). 19. Chopra, A.K. and Goel, R.K. A modal pushover analysis procedure to estimate seismic demands for unsymmetric-plan buildings", Earthq. Eng. Struct. Dyn., 33(8), pp. 903{927 (2004). 20. Chopra, A.K. and Goel, R.K. A modal pushover analysis procedure for estimating seismic demands for buildings", Earthquake Engineering and Structural Dynamics, 31(3), pp. 561{582 (2002). 21. Cheng, F.Y. Eigensolution techniques and undamped response analysis of multiple-degree-of-freedom systems", In Matrix Analysis of Structural Dynamics, pp. 48{98, Marcel Dekker, Rolla, Missouri, USA (2001). 22. Bathe, K.J. Eigen problems", In Finite Element Procedures in Engineering Analysis, pp. 838{979, Prentice Hall, New Jersey, USA (1996). 23. Newmark, N.M. A method of computation for structural dynamics", Journal of the Engineering Mechanics, 85(7), pp. 67{94 (1959). 24. Houbolt, J.C. A recurrence matrix solution for the dynamic response of elastic aircraft", J. Aeronaut. Sci., 17(9), pp. 540{550 (1950). 25. Wilson, E.L., Farhoomand, I., and Bathe, K.J. Nonlinear dynamic analysis of complex structures", Earthq. Eng. Struct. Dyn., 1(March 1972), pp. 241{ 252 (1973). 26. Bathe, K.J. and Wilson, E.L., Numerical Methods in Finite Element Analysis, pp. 1{528, Prentice Hall, New Jersey, USA (1976). 27. Bathe, K.J. and Wilson, E.L. Stability and accuracy analysis of direct integration methods" , Earthq. Eng. Struct. Dyn., 1(3), pp. 283{291 (1972). 28. Felippa, C.A. and Park, K.C. Direct time integration method in nonlinear structural dynamics", Comput. Method Appl. Mech. Eng., 17(18), pp. 277{313 (1979). 29. Johnson, D.E. A proof of the stability of the houbolt method", AIAA, 4(8), pp. 1450{1451 (1966). 30. Gladwell, I. and Thomas, R. Stability properties of the Newmark, Houbolt and Wilson methods", Int. J. Numer. Methods Anal. Methods Eng., 4(August 1979), pp. 143{158 (1980). 31. Park, K.C. An improved sti_y stable method for direct integration of nonlinear structural dynamic equations", J. Appl. Mech., 42(2), pp. 464{470 (1975). 32. Bathe, K.J. and Wilson, E.L. NONSAP - A nonlinear structural analysis program", Nucl. Eng. Des., 29(2), pp. 266{293 (1974). 33. Paultre, P. Direct time integration of linear systems", In Dynamics of Structures, pp. 223{248, John Wiley & Sons, Surrey, GBR (2013). 34. Acary, V. Energy conservation and dissipation properties of time-integration methods for nonsmooth elastodynamics with contact", Journal of Applied Mathematics and Mechanics, 96(5), pp. 585{603 (2016). M. Jalili Sadr Abad et al./Scientia Iranica, Transactions A: Civil Engineering 27 (2020) 1091{1112 1107 35. Fleming, A., Penesis, I., Macfarlane, G., Bose, N., and Denniss, T. Energy balance analysis for an oscillating water column wave energy converter", Ocean Engineering, 54, pp. 26{33 (2012). 36. Lee, T., Leok, M., and McClamroch, N.H. Geometric numerical integration for complex dynamics of tethered spacecraft", American Control Conference (ACC), San Francisco, California, USA, pp. 1885{1891 (2011). 37. Dowell, E.H. Nonlinear aeroelasticity", In A Modern Course in Aeroelasticity, pp. 487{529, Springer, NC, USA (2015). 38. Stanton, S.C., Erturk, A., Mann, B.P., Dowell, E.H., and Inman, D.J. Nonlinear nonconservative behavior and modeling of piezoelectric energy harvesters including proof mass e_ects", J. Intell. Mater. Syst. Struct., 23(2), pp. 183{199 (2012). 39. Hirsch, C. An Initial Guide to CFD", In Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid Dynamics, 2nd edition, pp. 1{12, Elsevier, Amesterdam, Netherlands (2007). 40. Yazdi, M.K. and Tehrani, P.H. The energy balance to nonlinear oscillations via Jacobi collocation method", Alexandria Eng. J., 54(2), pp. 99{103 (2015). 41. Mehdipour, I., Ganji, D.D., and Moza_ari, M. Application of the energy balance method to nonlinear vibrating equations", Current Applied Physics, 10(1), pp. 104{112 (2010). 42. Jalili Sadr Abad, M., Mahmoudi, M., and Dowell, E.H. Dynamic analysis of SDOF systems using modi_ed energy method", ASIAN J. Civ. Eng., 18(7), pp. 1125{ 1146 (2017). 43. Paz, M. and Leigh, W. Structures modeled as shear buildings", In Structural Dynamics, pp. 205{231, Springer, Boston, USA (2004). 44. Feynman, R.P., Leighton, R.B., and Sands, M., The Feynman Lectures on Physics, Desktop Edition, 1, Basic Books (2013). 45. Hall, K.C., Ekici, K., Thomas, J.P., and Dowell, E.H. Harmonic balance methods applied to computational uid dynamics problems", Int. J. Comut. Fluid Dyn., 27(2), pp. 52{67 (2013). 46. Light, J.C. and Walker, R.B. An R matrix approach to the solution of coupled equations for atom-molecule reactive scattering", J. Chem. Phys., 65(10), pp. 4272{ 4282 (1976). 47. Wang, Y., Ding, H., and Chen, L.Q. Asymptotic solutions of coupled equations of supercritically axially moving beam", Nonlinear Dyn., 87(1), pp. 25{36 (2016). 48. Armour, E.A.G. and Plummer, M. Calculation of the resonant contribution to Ze_(k0) using close-coupled equations for positron{molecule scattering", J. Phys. B At. Mol. Opt. Phys., 49(15), pp. 1{11 (2016). 49. Haojiang, D. General solutions for coupled equations for piezoelectric media", Int. J. Solids Struct., 33(16), pp. 2283{2298 (1996). 50. Polyanin, A.D. and Lychev, S.A. Decomposition methods for coupled 3D equations of applied mathematics and continuum mechanics: Partial survey, classi_cation, new results, and generalizations", Appl. Math. Model., 40(4), pp. 3298{3324 (2016). 51. Mourad, A. and Kamel, Z. An e_cient method for solving the MAS sti_ system of nonlinearly coupled equations: Application to the pseudoelastic response of shape memory alloys (SMA)", IOP Conf. Ser. Mater. Sci. Eng., 123(1), Miskolc-Lillafured, Hungary, pp. 1{ 5 (2016). 52. Nikitas, N., Macdonald, J.H.G., and Tsavdaridis, K.D. Modal analysis", Encycl. Earthq. Eng., 23, pp. 1{22 (2014). 53. Van Putten, M.H., Introduction to Methods of Approximation in Physics and Astronomy, Springer, Singapore (2017). 54. Duhamel, J.M.C., Elements de Calcul In_nitesimal, Mallet-Bachelier, France (1860) (In French). 55. Gavin, H.P., Classical Damping, Non-classical Damping, and Complex Modes, Dep. Civ. Environ. Eng. Duke Univ. NC, USA (2016). 56. Ghahari, S.F., Abazarsa, F., and Taciroglu, E. Blind modal identi_cation of non-classically damped structures under non-stationary excitations", Struct. Control Heal. Monit., 24(6), pp. 987{1006 (2017). 57. Cruz, C. and Miranda, E. Evaluation of the Rayleigh damping model for buildings", Eng. Struct., 138, pp. 324{336 (2017). 58. Bathe, K.J. Solution of equilibrium equations in dynamic analysis", In Finite Element Procedures, 2nd Edn., pp. 768{837, Prentice Hall, New Jersey, USA (2014). 59. Ben-Israel, A. A Newton-Raphson method for the solution of systems of equations", J. Math. Anal. Its Appl., 15, pp. 243{252 (1966). 60. Rheinboldt, W.C. Methods of Newton Type", In Methods for Solving Systems of Nonlinear Equations, 2nd Edn., pp. 35{43, Society for Industrial and Applied Mathematics (SIAM), Pittsburg, Pennsylvania, USA (1998).