Engineering Mechanics ›› 2018, Vol. 35 ›› Issue (10): 1-9,36.

### AN ENERGY CONSISTENT INTEGRATION METHOD FOR TRUSS ELEMENTS

PAN Tian-lin1,2, WU Bin2,3

1. 1. School of Civil Engineering and Architecture, Northeast Electric Power University, Jilin 132012, China;
2. School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China;
3. School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China
• Received:2017-06-06 Revised:2018-01-10 Online:2018-10-12 Published:2018-10-12

Abstract: Based on the energy equilibrium theory, an energy consistent integration method for truss elements is proposed in this paper. The method is unconditionally stable in nonlinear systems, and its accuracy is second order. The existence of algorithm parameters is proved by mean value theorem, and the solution form of the parameters is also provided. The discrete dynamic equations are linearized to obtain the equivalent stiffness matrices for iteration. The new algorithm is embedded in a nonlinear finite element program. On the basis of this program, the nonlinear dynamic analysis of a single pendulum and a transmission tower structure is completed. The numerical results show that the classic average acceleration method and implicit midpoint method are both energy inconsistent and may even produce divergent results. In contrast, the proposed method has good stability within different time steps.

CLC Number:

• TU311.4
  Chang S Y. Explicit pseudodynamic algorithm with unconditional stability[J]. Journal of Engineering Mechanics, 2002, 128(9):935-947. Chen C, Ricles J M. Stability analysis of SDOF real-time hybrid testing systems with explicit integration algorithms and actuator delay[J]. Earthquake Engineering & Structural Dynamics. 2008, 37(4):597-613. Chen C, Ricles J M, Marullo T M, et al. Real-time hybrid testing using the unconditionally stable explicit CR integration algorithm[J]. Earthquake Engineering & Structural Dynamics. 2009, 38(1):3-44. Hughes T J R. The finite element method:linear static and dynamic finite element analysis[M]. Massachusetts:Courier Corporation, 2000:551-553. Combescure D, Pegon P. α-Operator splitting time integration technique for pseudodynamic testing error propagation analysis[J]. Soil Dynamics and Earthquake Engineering, 1997, 16(7):427-443. Newmark N M. A method of computation for structural dynamics[J]. Journal of the Engineering Mechanics Division, 1959, 85(3):67-94. 周惠蒙, 吴斌, 王涛, 等. 基于速度的显式等效力控制方法的研究[J]. 工程力学, 2016, 33(6):15-22. Zhou Huimeng, Wu Bin, Wang Tao, et al. Explicit equivalent force control method based on velocity[J]. Engineering Mechanics, 2016, 33(6):15-22. (in Chinese) Hilber H M, Hughes T J R, Taylor R L. Improved numerical dissipation for time integration algorithms in structural dynamics[J]. Earthquake Engineering & Structural Dynamics, 1977, 5(3):283-292. Hilber H M, Hughes T J R. Collocation, dissipation and overshoot for time integration schemes in structural dynamics[J]. Earthquake Engineering & Structural Dynamics, 1978, 6(1):99-117. Wood W L, Bossak M, Zienkiewicz O C. An alpha modification of newmark's method[J]. International Journal for Numerical Methods in Engineering, 1980, 15(10):1562-1566. Chung J, Hulbert G M. A time integration algorithm for structural dynamics with improved numerical dissipation:the generalized-α method[J]. Journal of applied mechanics, 1993, 60(2):371-375. 梁轩, 杜建镔. 采用减震榫桥梁非线性动力学分析计算方法[J]. 工程力学, 2016, 33(4):136-143. Liang Xuan, Du Jianbin. An approach to nonlinear dynamic analysis of bridge with aseismic absorbers[J]. Engineering Mechanics, 2016, 33(4):136-143. (in Chinese) Kadapa C, Dettmer W G, Perić D. On the advantages of using the first-order generalised-alpha scheme for structural dynamic problems[J]. Computers & Structures, 2017, 193:226-238. Rossi S, Abboud N, Scovazzi G. Implicit finite incompressible elastodynamics with linear finite elements:A stabilized method in rate form[J]. Computer Methods in Applied Mechanics and Engineering, 2016, 311:208-249. Butcher J C. Numerical methods for ordinary differential equations, second edition[M]. Chichester:John Wiley & Sons Ltd. 2008:248-252. Simo J C, Hughes T J R. Computational inelasticity[M]. New York:Springer Science & Business Media, 1998:53-57. Li Y, Wu B, Ou J. Stability of average acceleration method for structures with nonlinear damping[J]. Earthquake Engineering and Engineering Vibration, 2006, 5(1):87-92. 潘天林, 吴斌. 隐式中点法对于非线性阻尼结构的稳定性[J]. 振动与冲击, 2013, 32(23):38-42. Pan Tianlin, Wu Bin. Stability of implicit midpoint algorithm applied to nonlinear damping structure[J]. Journal of Vibration and Shock, 2013, 32(23):38-42. (in Chinese) Labudde R A, Greenspan D. Discrete mechanics-A general treatment[J]. Journal of Computational Physics, 1974, 15(2):134-167. Hughes T J R, Caughey T K, Liu W K. Finite-element methods for nonlinear elastodynamics which conserve energy[J]. Journal of Applied Mechanics, 1978, 45(2):366-370. Kuhl D, Ramm E. Generalized energy-momentum method for non-linear adaptive shell dynamics[J]. Computer Methods in Applied Mechanics and Engineering, 1999, 178(3):343-366. Kuhl D, Crisfield M A. Energy-conserving and decaying algorithms in non-linear structural dynamics[J]. International Journal for Numerical Methods in Engineering, 1999, 45(5):569-599. Simo J C, Tarnow N. The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics[J]. Zeitschrift für angewandte Mathematik und Physik ZAMP, 1992, 43(5):757-792. Romero I. An analysis of the stress formula for energy-momentum methods in nonlinear elastodynamics[J]. Computational Mechanics, 2012, 50(5):603-610. Noels L, Stainier L, Ponthot J P. A first-order energy-dissipative momentum conserving scheme for elasto-plasticity using the variational updates formulation[J]. Computer Methods in Applied Mechanics and Engineering, 2008, 197(6):706-726. Meier C, Popp A, Wall W A. Geometrically exact finite element formulations for slender beams:Kirchhoff-love theory versus simo-reissner theory[J]. Archives of Computational Methods in Engineering, 2017, 24:1-81. Nguyen T L, Sansour C, Hjiaj M. Long-term stable time integration scheme for dynamic analysis of planar geometrically exact Timoshenko beams[J]. Journal of Sound and Vibration, 2017, 396:144-171. Crisfield M A, Shi J. A Co-rotational element time-integration strategy for non-linear dynamics[J]. International Journal for Numerical Methods in Engineering, 1994, 37(11):1897-1913. Crisfield M A, Shi J. An energy conserving co-rotational procedure for non-linear dynamics with finite elements[J]. Nonlinear Dynamics, 1996, 9(1/2):37-52. 潘天林. 能量一致积分方法及其在混合实验中的应用[D]. 哈尔滨:哈尔滨工业大学, 2016:20-27. Pan Tianlin. Energy consistent integration method and its applications to hybrid testing[D]. Harbin:Harbin Institute of Technology, 2016:20-27. (in Chinese) 潘天林, 吴斌, 郭丽娜, 等. 能量守恒逐步积分方法在工程结构动力分析中的应用[J]. 工程力学, 2014, 31(9):21-27. Pan Tianlin, Wu Bin, Guo Lina, et al. Application of energy conserving step-by-step integration algorithm in dynamic analysis of engineering structures[J]. Engineering Mechanics, 2014, 31(9):21-27. (in Chinese) Bathe K J, Baig M M I. On a composite implicit time integration procedure for nonlinear dynamics[J]. Computers & Structures, 2005, 83(31):2513-2524.
  WANG Tao, PAN Peng. STUDY AND APPLICATION OF SUBSTRUCTURE ONLINE HYBRID TEST METHOD [J]. Engineering Mechanics, 2018, 35(2): 1-12.  YUAN Fang, PANG Kun, DONG Cheng-ying, XU Zhi-jun. THE PFC3D NUMERICAL SIMULATION ON DYNAMIC PRESSURES AND FLOW OF SIDEDRAW SILOS [J]. Engineering Mechanics, 2016, 33(增刊): 301-305.  YANG Gang, FU Yi-ke, ZHENG Jian-min, HU De-an. SIMULATION OF LONG-ROD PROJECTILE PENETRATING REACTIVE ARMOR USING THE FE-SPH ADAPTIVE COUPLING METHOD [J]. Engineering Mechanics, 2016, 33(1): 223-231.  ZHU Ying, ZHANG Hong-tao, BAI Yu-xing. THE APPLICATION OF FINITE ELEMENT METHOD OF LINES WITH RECTANGULAR STRIP ELEMENT IN THE STEADY-STATE ANALYSIS OF THREE DIMENSIONAL THERMAL FIELD [J]. Engineering Mechanics, 2014, 31(增刊): 22-26.  WANG Xin-zheng,LI Ping,YANG Wen-xi,WANG Feng-chao,YUAN Xiao-ning. DAMAGE EVOLUTION OF CONCRETE FILLED STEEL TUBE BY LATERAL IMPACT WITH NUMERICAL SIMULATION [J]. Engineering Mechanics, 2013, 30(增刊): 267-272.  JIANG Nai-bin;LIU Zhan-fang. NUMERICAL ANALYSIS ON DYNAMIC RESPONSE OF RATE-DEPENDENT SATURATED POROUS MEDIA [J]. Engineering Mechanics, 2011, 28(9): 137-142.  CUI Jing-hao. SIGNIFICANCE OF MECHANICS IN DEVELOPMENT OF RELATED SUBJECTS AND NATIONAL ECONOMY [J]. Engineering Mechanics, 2010, 27(增刊Ⅱ): 1-041.  HUANG Li-yan;CUI Jing-hao;. IMPROVING TECHNICAL QUALITY OF THE JOURNAL BY HOLDING ACADEMIC CONFERENCES [J]. Engineering Mechanics, 2010, 27(增刊Ⅱ): 152-156.  LI Yan;WU Bin;OU Jin-ping;. REAL-TIME ENERGY-CONSERVING SUBSTRUCTURE TEST OF BUCKLING-RESTRAINED BRACE [J]. Engineering Mechanics, 2010, 27(增刊Ⅱ): 157-162.  YE Mao;TAN Ping;REN Min;ZHOU Fu-lin;WANG Dao-yuan. MODAL ANALYSIS OF MULTI-SPAN BEAMS WITH INTERMEDIATE FLEXIBLE CONSTRAINTS AND DIFFERENT BOUNDARY CONDITIONS [J]. Engineering Mechanics, 2010, 27(9): 80-085.  LI Ming;;WU Lin-zhi;GUAN Zheng-xi;MA Li;XIONG Jian. THE BUCKLING ANALYSIS OF SANDWICH COLUMNS REINFORCED BY METALLIC TUBES [J]. Engineering Mechanics, 2010, 27(12): 34-039.  LI Yan;WU Bin;OU Jin-ping;. SUBSTRUCTURE TESTING METHODS BASED ON ENERGY-CONSERVING TIME INTEGRATION ALGORITHM [J]. Engineering Mechanics, 2010, 27(1): 1-007.  ZHAO Wei-tao;ZHANG Da-qian;ZHANG Xu. A COUPLED METHOD OF STATIC STRENGTH AND FATIGUE RELIABILITY ANALYSIS OF TRUSS SYSTEM [J]. Engineering Mechanics, 2010, 27(1): 23-027,.  HUANG Li-yan;CUI Jing-hao;. DEVELOPMENT AND PROSPECT OF ‘ENGINEERING MECHANICS’ [J]. Engineering Mechanics, 2009, 26(10): 1-013.  LIU Tie-lin;JIANG Ying-chun;;LIU Hong-fei. DYNAMIC RESPONSE ANALYSES BY WAVE APPROACH ON HIGH-RISE STRUCTURES UNDER SEISMIC LOADING [J]. Engineering Mechanics, 2008, 25(增刊Ⅱ): 164-167.
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