Finite Element Analysis of Space Frame Structures

Chapter
Part of the Springer Series in Reliability Engineering book series (RELIABILITY)

Abstract

This chapter is devoted to the mechanics of space frame structures and presents necessary formulations for the finite element analysis of space frames. It contains eight sections. Section 1.1 is introduction which summarizes briefly the beam theories in general. Section 1.2 presents deformations, equilibrium equations and rotations, differential equations and solutions, stiffness and mass matrices, consistent load vectors of three-dimensional (3D) Timoshenko beams in general. It explains also the coordinate systems and transformations of 3D beams used in the FEA. Section 1.3 is devoted to member releases and formulation of partly connected members which are represented by spring-beam elements. Section 1.4 explains the formulation of eccentrically connected members. Section 1.5 presents formulations of an interface beam element to take into account soil–beam interactions in the analysis. It explains also soil deformations under Rayleigh wave propagation and calculation of the exerted forces. Section 1.6 is devoted to the calculation of natural frequencies and mode shapes. Consequently, Sect. 1.7 presents dynamic response analysis. Section 1.8 is devoted to examples.

Notes

Acknowledgments

With thanks to ASME for kindly granting permission for the reuse of materials printed in OMAE-2010 [73] as adopted in this chapter in Sect. 1.3.

References

  1. 1.
    Rutenbwg A (1979) Plane frame analysis of laterally loaded asymmetric buildings-an uncoupled solution. Comp Struct 10(3):553–555CrossRefGoogle Scholar
  2. 2.
    Wilkinson S, Thambiratnam D (2001) Simplified procedure for seismic analysis of asymmetric buildings. Comp Struct 79(32):2833–2845CrossRefGoogle Scholar
  3. 3.
    Kam TY, Lee FS (1986) Nonlinear analysis of steel plane frames with initial imperfections. Comp Struct 23(4):553–557CrossRefGoogle Scholar
  4. 4.
    Sophianopoulos DS (2003) The effect of joint flexibility on the free elastic vibration characteristics of steel plane frames. J Constr Steel Res 59(8):995–1008CrossRefGoogle Scholar
  5. 5.
    Blandford GE (1988) Static analysis of flexibly connected thin-walled plane frames. Comp Struct 28(1):105–113MATHCrossRefGoogle Scholar
  6. 6.
    Dissanayake UI, Burgess IW, Davison JB (2000) Modelling of plane composite frames in unpropped construction. Eng Struct 22(4):287–303CrossRefGoogle Scholar
  7. 7.
    Sekulovic M, Salatic R, Nefovska M (2002) Dynamic analysis of steel frames with flexible connections. Comp Struct 80(11):935–955CrossRefGoogle Scholar
  8. 8.
    Lui EM, Chen WF (1986) Analysis and behaviour of flexibly-jointed frames. Eng Struct 8(2):107–118CrossRefGoogle Scholar
  9. 9.
    Lee HP, Ng TY (1994) In-plane vibration of planar frame structures. J Sound Vib 172(3):420–427MATHCrossRefGoogle Scholar
  10. 10.
    Labuschagne A, van Rensburg NFJ, van der Merwe AJ (2009) Comparison of linear beam theories. J Math Comput Modell 49(1–2):20–30MATHCrossRefGoogle Scholar
  11. 11.
    The LH (2004) Beam element verification for 3D elastic steel frame analysis. Comput Struct 82(15–16):1167–1179Google Scholar
  12. 12.
    Kim SE, Kim Y, Choi SH (2001) Nonlinear analysis of 3-D steel frames. Thin Walled Struct 39(6):445–461CrossRefGoogle Scholar
  13. 13.
    Kumar P, Nukala VV, White DW (2004) A mixed finite element for three-dimensional nonlinear analysis of steel frames. Comput Meth Appl Mech Eng 193(23–26):2507–2545MATHGoogle Scholar
  14. 14.
    Yang YB (1993) Recent researches on buckling of framed structures and curved beams. J Constr Steel Res 26(2–3):193–210CrossRefGoogle Scholar
  15. 15.
    Palaninathan R, Chandrasekharan PS (1985) Curved beam element stiffness matrix formulation. Comput Struct 21(4):663–669MATHCrossRefGoogle Scholar
  16. 16.
    Choit J, Lim J (1995) General curved beam elements based on the assumed strain fields. Comput Struct 55(3):379–386CrossRefGoogle Scholar
  17. 17.
    Yang SY, Sin HC (1995) Curvature-based beam elements for the analysis of Timoshenko and shear-deformable curved beams. J Sound Vib 187(4):569–584MATHCrossRefGoogle Scholar
  18. 18.
    Culver CG, Oestel DJ (1969) Natural frequencies of multispan curved beams. J Sound Vib 10(3):380–389CrossRefGoogle Scholar
  19. 19.
    Yoona KY, Parkb NH, Choic YJ, Kangd YJ (2006) Natural frequencies of thin-walled curved beams. Finite Elem Anal Des 42(13):1176–1186CrossRefGoogle Scholar
  20. 20.
    Dym CL, Shames IH (1973) Solid mechanics: a variational approach. McGraw·Hill, New York, pp 175–213Google Scholar
  21. 21.
    Boresi AP, Schmidt RJ (2003) Advanced mechanics of materials, 6th edn. Wiley, New YorkGoogle Scholar
  22. 22.
    Case J, Chilver L, Ross CTF (1999) Strength of materials and structures, 4th edn. Elsevier, AmsterdamGoogle Scholar
  23. 23.
    Love AEH (1963) A treatise on the mathematical theory of elasticity, 4th edn. Cambridge University Press, New YorkGoogle Scholar
  24. 24.
    Timoshenko SP, Goodier JN (1969) Theory of elasticity, 3rd edn. McGraw-Hill, NewYorkGoogle Scholar
  25. 25.
    Ruge P, Birk C (2007) A comparison of infinite Timoshenko and Euler-Bernoulli beam models on Winkler foundation in the frequency- and time-domain. J Sound Vib 304(3–5):932–947CrossRefGoogle Scholar
  26. 26.
    Ganesan N, Engels RC (1992) Hierarchical Bernoulli-Euler beam finite elements. Comput Struct 43(2):297–304MATHCrossRefGoogle Scholar
  27. 27.
    Timoshenko SP (1921) On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos Mag 41(245):744–746CrossRefGoogle Scholar
  28. 28.
    Timoshenko SP (1922) On the transverse vibrations of bars of uniform cross-section. Philos Mag 43(253):125–131CrossRefGoogle Scholar
  29. 29.
    Wang CM (1995) Timoshenko beam-bending solutions in terms of Euler–Bernoulli solutions. J Eng Mech 121(6):763–765CrossRefGoogle Scholar
  30. 30.
    Davis R, Henshell RD, Warburton GB (1972) A Timoshenko beam element. J Sound Vib 22(4):475–487CrossRefGoogle Scholar
  31. 31.
    Bakr EM, Shabana AA (1987) Timoshenko beams and flexible multibody system dynamics. J Sound Vib 116(1):89–107MATHCrossRefGoogle Scholar
  32. 32.
    Thomas DL, Wilson JM, Wilson RR (1973) Timoshenko beam finite elements. J Sound Vib 31(3):315–330MATHCrossRefGoogle Scholar
  33. 33.
    Horr AM, Schmidt LC (1995) Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package. Comput Struct 55(3):405–412MATHCrossRefGoogle Scholar
  34. 34.
    Bazoune A, Khulief YA, Stephen NG (2003) Shape functions of three-dimensional Timoshenko beam element. J Sound Vib 259(2):473–480CrossRefGoogle Scholar
  35. 35.
    Lou P, Dai GL, Zeng QY (2006) Finite-element analysis for a Timoshenko beam subjected to a moving mass. Proc Inst Mech Eng Part C J Mech Eng Sci 220(5):669–678CrossRefGoogle Scholar
  36. 36.
    Yu W, Hodges DH (2005) Generalized Timoshenko theory of the variational asymptotic beam sectional analysis. J Am Helicopter Soc 50(1):46–55CrossRefGoogle Scholar
  37. 37.
    Mikkola A, Dmitrochenko O, Matikainen M (2009) Inclusion of transverse shear deformation in a beam element based on the absolute nodal coordinate formulation. J Comput Nonlinear Dyn 4(1):011004–011012CrossRefGoogle Scholar
  38. 38.
    Ghali A, Neville AM, Brown TG (2009) Structural analysis: a unified classical and matrix approach, 6th edn. Spon, LondonGoogle Scholar
  39. 39.
    McCormac CJ (2007) Structural analysis: using classical and matrix methods, 4th edn. Wiley, HobokenGoogle Scholar
  40. 40.
    Kurrer KE (2008) The history of the theory of structures: from arch analysis to computational mechanics. Ernst & Sohn, BerlinGoogle Scholar
  41. 41.
    Przemieniecki JS (1968) Theory of matrix structural analysis. McGraw-Hill, New YorkMATHGoogle Scholar
  42. 42.
    Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics, 6th edn. Elsevier Butterworth-Heinemann, OxfordMATHGoogle Scholar
  43. 43.
    Crisfield MA (1986) Finite elements and solution procedures for structural analysis: linear analysis, vol 1. Pineridge Press, SwanseaGoogle Scholar
  44. 44.
    Wunderlich W, Pilkey WD (2003) Mechanics of structures: variational and computational methods, 2nd edn. CRC Press, Boca RatonMATHGoogle Scholar
  45. 45.
    Slivker VI (2007) Mechanics of structural elements: theory and applications. Springer, BerlinCrossRefGoogle Scholar
  46. 46.
    Brenner SC, Scott LR (2008) The mathematical theory of finite element methods, 3rd edn. Springer, New YorkMATHCrossRefGoogle Scholar
  47. 47.
    Chen Z (2005) Finite element methods and their applications. Springer, BerlinMATHGoogle Scholar
  48. 48.
    Ross CTF (1999) Advanced finite element methods. Horwood, ChichesterGoogle Scholar
  49. 49.
    Bathe KJ, Wilson EL (1976) Numerical methods in finite element analysis. Prentice-Hall, EnglewoodMATHGoogle Scholar
  50. 50.
    Omid BN, Rankin CC (1991) Finite rotation analysis and consistent linearization using projectors. Comput Meth Appl Mech Eng 93(3):353–384MATHCrossRefGoogle Scholar
  51. 51.
    Ibrahimbegovic A (1997) On the choice of finite rotation parameters. Comput Meth Appl Mech Eng 149:49–71MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Li M (1998) The finite deformation theory for beam, plate and shell Part III. The three-dimensional beam theory and the FE formulation. Comput Meth Appl Mech Eng 162(1–4):287–300MATHCrossRefGoogle Scholar
  53. 53.
    Kim MY, Chang SP, Park HG (2001) Spatial postbuckling analysis of nonsymmetric thin-walled frames. I: theoretical considerations based on semitangential property. J Eng Mech 127(8):769–778CrossRefGoogle Scholar
  54. 54.
    Boresi AP, Chong KP (2000) Elasticity in engineering mechanics. Wiley, New YorkGoogle Scholar
  55. 55.
    Schramm U, Kitis L, Kang W, Pilkey WD (1994) On the shear deformation coefficient in beam theory. Finite Elem Anal Des 16:141–162MATHCrossRefGoogle Scholar
  56. 56.
    Pilkey WD (2002) Analysis and design of elastic beams, computational methods. Wiley, New YorkCrossRefGoogle Scholar
  57. 57.
    Clough RW, Penzien J (1993) Dynamics of structures, 2nd edn. McGraw-Hill, New YorkGoogle Scholar
  58. 58.
    Bishop RED, Gladwell GML, Michaelson S (1965) The matrix analysis of vibration. Cambridge University Press, LondonMATHGoogle Scholar
  59. 59.
    Meirovitch L (1967) Analytical methods in vibrations. Collier-McMillan Limited, LondonMATHGoogle Scholar
  60. 60.
    Woodhouse J (1998) Linear damping models for structural vibration. J Sound Vib 215(3):547–569MATHCrossRefGoogle Scholar
  61. 61.
    Liang Z, Lee GC (1991) Representation of damping matrix. J Eng Mech 117(5):1005–1020CrossRefGoogle Scholar
  62. 62.
    Hinton E, Rock T, Zienkiewicz OC (1976) A note on mass lumping and related processes. J Earthq Eng Struct Dyn 4:245–249CrossRefGoogle Scholar
  63. 63.
    Barltrop NDP, Adams AJ (1991) Dynamics of fixed marine structures, 3rd edn. Butterworth-Heinemann Ltd., Oxford, pp 41–48Google Scholar
  64. 64.
    Chen B, Hu Y, Tan M (1990) Local joint flexibility of tubular joints of offshore structures. Mar Struct 3(3):177–197CrossRefGoogle Scholar
  65. 65.
    Bouwkamp JG, Hollings JP, Malson BF, Row DG (1980) Effect of joint flexibility on the response of offshore structures. Proceedings of the offshore technology conference, Texas, USAGoogle Scholar
  66. 66.
    Fessler H, Mockford PB, Webster JJ (1986) Parametric equations for the flexibility matrices of single brace tubular joint in offshore structures. Proc Inst Civ Eng Part 2 81:659–673CrossRefGoogle Scholar
  67. 67.
    Fessler H, Mockford PB, Webster JJ (1986) Parametric equations for the flexibility matrices of multi-brace tubular joint in offshore structures. Proc Inst Civ Eng Part 2 81:675–696CrossRefGoogle Scholar
  68. 68.
    Ueda Y, Rashed SMH, Nakacho K (1987) An improved joint model and equations for flexibility of tubular joints. Proceedings of the 6th international symposium offshore Mechanics and Arctic Engineering, Texas, USAGoogle Scholar
  69. 69.
    Lee DG, Song YH (1993) Efficient seismic analysis of piping systems with joint deformations. Eng Struct 15(1):2–12CrossRefGoogle Scholar
  70. 70.
    Hu Y, Chen B, Ma J (1993) An equivalent element representing local flexibility of tubular joints in structural analysis of offshore platforms. Comput Struct 47(6):957–969CrossRefGoogle Scholar
  71. 71.
    Chen TY, Zhang HY (1996) Stress analysis of spatial frames with consideration of local flexibility of multiplanar tubular joint. Eng Struct 18(6):465–471CrossRefGoogle Scholar
  72. 72.
    Karadeniz H (1994) An algorithm for member releases and partly connected members in offshore structural analysis. Proceedings of the 13th international conference on Offshore Mechanics and Arctic Engineering, OMAE-94, Texas, USA, vol 1, pp 471–476Google Scholar
  73. 73.
    Karadeniz H (2010) A calculation model for deteriorated members of 3D frame structures in the static and dynamic analyses. Proceedings of the 29th international conference on Offshore Mechanics and Arctic Engineering, OMAE-2010, Paper No. OMAE2010-20971, Shanghai, ChinaGoogle Scholar
  74. 74.
    Baigent AH, Hancock GJ (1982) Structural analysis of assemblages of thin-walled members. Eng Struct 4:207–216CrossRefGoogle Scholar
  75. 75.
    Faruque MO (1986) Mechanics of material interfaces. In: Selvadurai APS, Voyiadjis GS (eds) An axisymmetric interface element for soil-structure interaction problems. Elsevier, AmsterdamGoogle Scholar
  76. 76.
    Boulon M, Garnica P, Vermeer PA (1995) Mechanics of geomaterial interfaces. In: Selvadurai APS, Boulon MJ (eds) Soil-structure interaction: FEM computations. Elsevier, AmsterdamGoogle Scholar
  77. 77.
    Selvadurai APS (1995) Mechanics of geomaterial interfaces. In: Selvadurai APS, Boulon MJ (eds) Boundary element modeling of geomaterial interfaces. Elsevier, AmsterdamGoogle Scholar
  78. 78.
    Karadeniz H (2001) Earthquake analysis of buried structures and pipelines based on Rayleigh wave propagation. Int J Offshore Polar Eng 11(2):133–140Google Scholar
  79. 79.
    Miranda C, Nair K (1966) Finite beams on elastic foundation. J Struct Eng 92(ST2):131–141Google Scholar
  80. 80.
    Selvadurai APS (1979) Elastic analysis of soil-foundation interaction. Elsevier, AmsterdamGoogle Scholar
  81. 81.
    Ting BY (1982) Finite beams on elastic foundation with restrains. J Struct Eng 108(ST3):611–621Google Scholar
  82. 82.
    Ting BY, Mockry EF (1984) Beam on elastic foundation finite element. J Struct Eng 110(10):2324–2339CrossRefGoogle Scholar
  83. 83.
    Aydogan M (1995) Stiffness-matrix formulation of beams with shear effect on elastic foundation. J Struct Eng 121(99):1265–1270Google Scholar
  84. 84.
    Yokoyama T (1991) Vibrations of Timoshenko beam-columns on two-parameter elastic foundations. Earthquake Eng Struct Dyn 20:353–370MathSciNetCrossRefGoogle Scholar
  85. 85.
    Ishihara K (1996) Soil behaviour in earthquake geotechnics. Oxford University Press, OxfordGoogle Scholar
  86. 86.
    Toki K, Sato T, Miura F (1981) Separation and sliding between soil and structure during strong ground motion. Earthquake Eng Struct Dyn 9:263–277CrossRefGoogle Scholar
  87. 87.
    Lay T, Wallace TC (1995) Modern global seismology. Academic Press, LondonGoogle Scholar
  88. 88.
    Takeuchi H, Saito M (1972) Seismic surface waves. In: Bold BA (ed) Methods in computational physics, seismology: surface waves and earth oscillations. Academic Press, LondonGoogle Scholar
  89. 89.
    Achenbach JD (1980) Wave propagation in elastic solids. North-Holland, AmsterdamGoogle Scholar
  90. 90.
    Graff KF (1975) Wave motion in elastic solids. Oxford University Press, LondonMATHGoogle Scholar
  91. 91.
    Makris N (1994) Soil-pile interaction during the passage of Rayleigh waves: an analytical solution. Earthq Eng Struct Dyn 23:153–167CrossRefGoogle Scholar
  92. 92.
    Wilkinson JH (1965) The algebraic eigenvalue problem. Clarendon Press, OxfordMATHGoogle Scholar
  93. 93.
    Jennings A (1977) Matrix computation for engineers and scientists. Wiley, LondonMATHGoogle Scholar
  94. 94.
    Parlett BN (1980) The symmetric eigenvalue problem. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
  95. 95.
    Bathe KJ, Wilson EL (1976) Finite element procedures in engineering analysis. Prentice-Hall, Englewood CliffsGoogle Scholar
  96. 96.
    Hudges TJR (1987) The finite element method, linear static and dynamic finite element analysis. Prentice-Hall, Englewood CliffsGoogle Scholar
  97. 97.
    Qian Y, Dhatt G (1995) An accelerated subspace method for generalized eigenproblems. Comput Struct 54(6):1127–1134MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    Karadeniz H (1996) An accelerated solution algorithm for the eigenvalue problem in the offshore structural analysis. Proc of ISOPE-96, 26–31 May, Los-Angeles, USAGoogle Scholar
  99. 99.
    Yamazaki F, Shinozuka M, Dasgupta G (1988) Neumann expansion for stochastic finite element analysis. J Eng Mech 114(8):1335–1354CrossRefGoogle Scholar
  100. 100.
    Dokainish MA, Subraraj K (1989) A survey of direct time-integration methods in computational structural dynamics-I: explicit methods. Comput Stuct 32(6):1371–1386MATHCrossRefGoogle Scholar
  101. 101.
    Subraraj K, Dokainish MA (1989) A survey of direct time-integration methods in computational structural dynamics-II: implicit methods. Comput Stuct 32(6):1387–1401CrossRefGoogle Scholar
  102. 102.
    Biggs JM (1964) Introduction to structural dynamics. McGraw-Hill, New YorkGoogle Scholar
  103. 103.
    Humar JL (2002) Dynamics of structures, 2nd edn. Swets & Zeitlinger B.V., LisseMATHGoogle Scholar
  104. 104.
    Karadeniz H (2009) SAPOS, spectral analysis program of structures. Report, Structural Mechanics Division, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The NetherlandsGoogle Scholar
  105. 105.
    Young WC, Budynas RG (2002) Roark’s formulas for stress and strain, 7th edn. McGraw-Hill, New YorkGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Halil Karadeniz
    • 1
  • Mehmet Polat Saka
    • 2
  • Vedat Togan
    • 3
  1. 1.Faculty of Civil Engineering and GeosciencesDelft University of TechnologyDelftThe Netherlands
  2. 2.Department of Engineering SciencesMiddle East Technical UniversityAnkaraTurkey
  3. 3.Department of Civil EngineeringKaradeniz Technical UniversityTrabzonTurkey

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