International Journal of Automotive Technology

, Volume 19, Issue 6, pp 1033–1040 | Cite as

Variable Cross-Section Rectangular Beam and Sensitivity Analysis for Lightweight Design of Bus Frame

  • Wenjie ZuoEmail author
  • Jiaxin Fang
  • Minghui Zhong
  • Guikai Guo


Timoshenko beam element of variable cross-section rectangular tube is developed and applied in the lightweight design of bus frame in this paper. Firstly, the finite element formulations of variable cross-section beam (VCB) are derived under the loadsteps of axial deformation, torsional deformation and bending deformation. Secondly, bending deformation experiment and its detailed shell finite element model (FEM) simulation of variable cross-section rectangular tube were conducted; and the proposed VCB, detailed shell FEM and experimental results can be highly consistent. Thirdly, VCBs are used to substitute for parts of the uniform ones in a bus frame. An innovatively lightweight bus frame is obtained and all the performance responses are improved simultaneously. Finally, rollover analysis further shows the advantage of variable cross-section bus frame in crashworthiness design.

Key Words

Variable cross-section beam Beam element Lightweight design Bus frame 


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  1. Arbabi, F. and Li, F. (1991). Buckling of variable crosssection columns: Integral-equation approach. J. Structural Engineering 117, 8, 2426–2441.CrossRefGoogle Scholar
  2. Bai, J., Li, Y. and Zuo, W. (2017). Cross-sectional shape optimisation for thin-walled beam crashworthiness with stamping constraints using genetic algorithm. Int. J. Vehicle Design 73, 1–3, 76−95.CrossRefGoogle Scholar
  3. Balesdent, M. and Chriette, A. (2012). A survey of multidisciplinary design optimization methods in launch vehicle design. Structural and Multidisciplinary Optimization 45, 5, 619–642.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Chen, W. and Zuo, W. (2014). Component sensitivity analysis of conceptual vehicle body for lightweight design under static and dynamic stiffness demands. Int. J. Vehicle Design 66, 2, 107–123.CrossRefGoogle Scholar
  5. Choi, I. S., Jang, G. W., Choi, S., Shin, D. and Kim, Y. Y. (2016). Higher order analysis of thin-walled beams with axially varying quadrilateral cross sections. Computers & Structures, 179, 127–139.CrossRefGoogle Scholar
  6. Eisenberger, M. (1990). Exact static and dynamic stiffness matrices for general variable cross section members. AIAA Journal 28, 6, 1105–1109.CrossRefzbMATHGoogle Scholar
  7. Eisenberger, M. (1991). Buckling loads for variable crosssection members with variable axial forces. Int. J. Solids and Structures 27, 2, 135–143.CrossRefGoogle Scholar
  8. Eisenberger, M. (1995). Dynamic stiffness matrix for variable cross-section Timoshenko beams. Communications in Numerical Methods in Engineering 11, 6, 507–513.CrossRefzbMATHGoogle Scholar
  9. Gaines, J. and Volterra, E. (1966). Transverse vibrations of cantilever bars of variable cross section. J. Acoustical Society of America 39, 4, 674–679.CrossRefGoogle Scholar
  10. Kim, H. and Jang, G. W. (2017). Higher-order thin-walled beam analysis for axially varying generally shaped cross sections with straight cross-section edges. Computers & Structures, 189, 83–100.CrossRefGoogle Scholar
  11. Kim, J. H. and Kim, Y. Y. (1999). Analysis of thin-walled closed beams with general quadrilateral cross sections. J. Applied Mechanics 66, 4, 904–912.CrossRefGoogle Scholar
  12. Kim, J. H. and Kim, Y. Y. (2000). One-dimensional analysis of thin-walled closed beams having general cross-sections. Int. J. Numerical Methods in Engineering 49, 5, 653–668.CrossRefzbMATHGoogle Scholar
  13. Lyu, N., Lee, B. and Saitou, K. (2006). Optimal subassembly partitioning of space frame structures for in-process dimensional adjustability and stiffness. J. Mechanical Design 128, 3, 527–535.CrossRefGoogle Scholar
  14. Lyu, N. and Saitou, K. (2005). Topology optimization of multicomponent beam structure via decompositionbased assembly synthesis. J. Mechanical Design 127, 2, 170–183.CrossRefGoogle Scholar
  15. Mayyas, A., Shen, Q., Mayyas, A., Abdelhamid, M., Shan, D., Qattawi, A. and Omar, M. (2011). Using quality function deployment and analytical hierarchy process for material selection of body-in-white. Materials & Design 32, 5, 2771–2782.CrossRefGoogle Scholar
  16. Ronagh, H., Bradford, M. and Attard, M. (2000a). Nonlinear analysis of thin-walled members of variable cross-section. Part I: Theory. Computers & Structures 77, 3, 285–299.Google Scholar
  17. Ronagh, H., Bradford, M. and Attard, M. (2000b). Nonlinear analysis of thin-walled members of variable cross-section. Part II: Application. Computers & Structures 77, 3, 301–313.Google Scholar
  18. Takezawa, A., Nishiwaki, S., Izui, K. and Yoshimura, M. (2007). Structural optimization based on topology optimization techniques using frame elements considering cross-sectional properties. Structural and Multidisciplinary Optimization 34, 1, 41–60.CrossRefGoogle Scholar
  19. Thomas, H., Zhou, M. and Schramm, U. (2002). Issues of commercial optimization software development. Structural and Multidisciplinary Optimization 23, 2, 97–110.CrossRefGoogle Scholar
  20. Torstenfelt, B. and Klarbring, A. (2007). Conceptual optimal design of modular car product families using simultaneous size, shape and topology optimization. Finite Elements in Analysis and Design 43, 14, 1050–1061.MathSciNetCrossRefGoogle Scholar
  21. Wang, H., Li, E., Li, G. and Zhong, Z. (2008a). Optimization of sheet metal forming processes by the use of space mapping based metamodeling method. Int. J. Advanced Manufacturing Technology 39, 7, 642–655.Google Scholar
  22. Wang, H., Li, G. and Zhong, Z. (2008b). Optimization of sheet metal forming processes by adaptive response surface based on intelligent sampling method. J. Materials Processing Technology 197, 1–3, 77−88.CrossRefGoogle Scholar
  23. Zou, M., Wei, C., Li, J., Xu, S. and Zhang, X. (2015). The energy absorption of bamboo under dynamic axial loading. Thin-Walled Structures, 95, 255–261.CrossRefGoogle Scholar
  24. Zou, M., Xu, S., Wei, C., Wang, H. and Liu, Z. (2016). A bionic method for the crashworthiness design of thinwalled structures inspired by bamboo. Thin-Walled Structures, 101, 222–230.CrossRefGoogle Scholar
  25. Zuo, W. (2013). An object-oriented graphics interface design and optimization software for cross-sectional shape of automobile body. Advances in Engineering Software, 64, 1–10.CrossRefGoogle Scholar
  26. Zuo, W. (2015). Bi-level optimization for the crosssectional shape of a thin-walled car body frame with static stiffness and dynamic frequency stiffness constraints. Proc. Institution of Mechanical Engineers, Part D: J. Automobile Engineering 229, 8, 1046–1059.Google Scholar
  27. Zuo, W. and Bai, J. (2016). Cross-sectional shape design and optimization of automotive body with stamping constraints. Int. J. Automotive Technology 17, 6, 1003–1011.CrossRefGoogle Scholar
  28. Zuo, W., Bai, J. and Li, B. (2014). A hybrid OC–GA approach for fast and global truss optimization with frequency constraints. Applied Soft Computing 14, Part C, 528−535.Google Scholar
  29. Zuo, W., Li, W., Xu, T., Xuan, S. and Na, J. (2012). A complete development process of finite element software for body-in-white structure with semi-rigid beams in.NET framework. Advances in Engineering Software 45, 1, 261–271.CrossRefGoogle Scholar
  30. Zuo, W. and Saitou, K. (2017). Multi-material topology optimization using ordered SIMP interpolation. Structural and Multidisciplinary Optimization 55, 2, 477–491.MathSciNetCrossRefGoogle Scholar
  31. Zuo, W., Yu, J. and Saitou, K. (2016). Stress sensitivity analysis and optimization of automobile body frame consisting of rectangular tubes. Int. J. Automotive Technology 17, 5, 843–851.CrossRefGoogle Scholar

Copyright information

© The Korean Society of Automotive Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Wenjie Zuo
    • 1
    • 2
    Email author
  • Jiaxin Fang
    • 2
  • Minghui Zhong
    • 1
  • Guikai Guo
    • 2
  1. 1.State Key Laboratory of Automobile Simulation and ControlJilin UniversityChangchunChina
  2. 2.School of Mechanical Science and EngineeringJilin UniversityChangchunChina

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