Engineering with Computers

, Volume 27, Issue 3, pp 201–210 | Cite as

CAD-integrated analysis of 3-D beams: a surface-integration approach

Original Article


Most engineering artifacts are designed and analyzed today within a 3-D computer aided design (CAD) environment. However, slender objects such as beams are designed in a 3-D environment, but analyzed using a 1-D beam-element, since their 3-D analysis exhibits locking and/or is computationally inefficient. This process is tedious and error-prone. Here, we propose a dual-representation strategy for designing and analyzing 3-D beams, directly within a 3-D CAD environment. The proposed method exploits classic 1-D beam physics, but is implemented within a 3-D CAD environment by appealing to the divergence theorem. Consequently, the proposed method is numerically and computationally equivalent to classic 1-D beam analysis for uniform cross-section beams. But, more importantly, it closely matches the accuracy of a full-blown 3-D finite element analysis for non-uniform beams.


Beams FEA Euler–Bernoulli Timoshenko 



The authors wish to acknowledge the support of the National Science Foundation under grants OCI-0636206, and CMMI-0726635, CMMI-0745398.


  1. 1.
    Requicha AG (1980) Representations for rigid solids: theory, methods, and systems. ACM Comput Surv (CSUR) 12(4):437–464CrossRefGoogle Scholar
  2. 2.
    Fu MW, Ong SK, Lu WF, Lee IBH, Nee AYC (2003) An approach to identify design and manufacturing features from a data exchanged part model. Comput Aided Des 35:979–993CrossRefGoogle Scholar
  3. 3.
    Pratt MJ, Anderson BD, Ranger T (2005) Towards the standardized exchange of parameterized feature-based CAD models. Comput Aided Des 37:1251–1265CrossRefGoogle Scholar
  4. 4.
    Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element method: its basis and fundamentals, 6th edn. Elsevier Butterworth Heinemann, AmsterdamGoogle Scholar
  5. 5.
    Dow J, Byrd DE (1988) The Identification and elimination of artificial stiffening errors in finite elements. Int J Numer Methods Eng 26(3):743–762MATHCrossRefGoogle Scholar
  6. 6.
    Jog CS (2005) A 27-node hybrid brick and a 21-node hybrid wedge element for structural analysis. Finite Elem Anal Des 41(11–12):1209–1232CrossRefGoogle Scholar
  7. 7.
    Duster A, Broker H, Rank E (2001) The p-version of finite element method for three-dimensional curved thin walled structures. Int J Numer Methods Eng 52(7):673–703CrossRefGoogle Scholar
  8. 8.
    Braess D, Kaltenbacher M (2007) Efficient 3D-finite-element-formulation for thin mechanical and piezoelectric structures. Int J Numer Methods Eng 73:147–161MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dorfmann A, Nelson RB (1995) Three-dimensional finite element for analysing thin plate/shell structures. Int J Numer Methods Eng 38(20):3453–3482MATHCrossRefGoogle Scholar
  10. 10.
    Jorabchi K, Danczyk J, Suresh K (2008) Algebraic reduction of beams for CAD-integrated analysis. CAD (submitted)Google Scholar
  11. 11.
    Wang CM, Reddy JN, Lee KH (2000) Shear deformable beams and plates: relationship to classical solutions. Elsevier Science, LondonGoogle Scholar
  12. 12.
    Pilkey W (2002) Analysis and Design of Elastic Beams. Wiley, New YorkCrossRefGoogle Scholar
  13. 13.
    Zhou D, Cheung YK (2001) Vibrations of tapered Timoshenko beams in terms of static Timoshenko beam elements. J Appl Mech 68(4):596–603MATHCrossRefGoogle Scholar
  14. 14.
    Lobontiu N, Garcia E (2004) Two microcantilever designs: lumped-parameter model for static and modal analysis. J Microelectromech Syst 13(1):41–50CrossRefGoogle Scholar
  15. 15.
    Lu ZR, Huang M, Liu JK, Chen WH, Liao WY (2009) Vibration analysis of multi-stepped beams with the composite element models. J Sound Vib 322:1070–1080CrossRefGoogle Scholar
  16. 16.
    Bronshtein IN, Semendyayev KA (1985) Handbook of mathematics. Van Nostrand Reinhold, New YorkGoogle Scholar
  17. 17.
    Rathod HT, Govinda Rao HS (1998) Integration of trivariate polynomials over linear polyhedra in euclidean three-dimensional space. J Aust Math Soc 39:355–385MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Tessler A, Dong SB (1981) On a hierarchy of conforming Timoshenko beam elements. Comput Struct 14(3–4):335–344CrossRefGoogle Scholar
  19. 19.
    SolidWorks (2005) SolidWorks.
  20. 20.
    Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics, 6th edn. Elsevier, OxfordMATHGoogle Scholar
  21. 21.
    Young WC (1989) Roark’s formulas for stress and strain. McGraw Hill, New YorkGoogle Scholar
  22. 22.
    Hsu J-C, Lee H-L, Chang W-J (2007) Flexural vibration frequency of atomic force microscopy cantilevers using the Timoshenko beam model. Nanotechnology 18:285503CrossRefGoogle Scholar
  23. 23.
    Akinpelu FO (2007) The effect of an attached mass on an Euler-Bernoulli beam. J Eng Appl Sci 2:1251–1254Google Scholar

Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  1. 1.2059, Mechanical Engineering BuildingUniversity of WisconsinMadisonUSA

Personalised recommendations