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First-and second-order sensitivities of beams with respect to cross-sectional cracks

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Summary

This paper deals with a variational formulation for the sensitivity problem of beam systems in the context of deformable solids with cracks. Natural frequencies are defined as state variables involved in the energy functional of the system, while the crack's length and position are treated as system parameters. The hierarchical equation system is formed and solved for the first and second derivatives of the natural frequency functions with respect to the crack's length and position. An analytical procedure for calculations of the second-order sensitivities of natural frequencies is proposed for the non-symmetrical equation system operator. Numerical algorithms are worked out and implemented computationally. Analytical and numerical aspects of the problem are discussed in detail through a number of illustrative results.

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References

  1. Collins, J.D.;Thomson, W.T.: The eigenvalue problem for structural systems with statistical properties. AIAA 7(4) (1969) 642–648

    MATH  Google Scholar 

  2. Dems, K.;Mroz, Z.: Variational approach by means of adjoint systems to structural optimization and sensitivity analysis. I. Variation of material parameters within a fixes domain. Solids Struct 19 (1983) 677–692

    Article  MATH  MathSciNet  Google Scholar 

  3. Drewko, J.: Elastic hinge modelling in vibration analysis of beams with cross-sections weakened by cracks. Marine Tech Trans 10 (1999) 93–103

    Google Scholar 

  4. Drewko, J.: On the models of elastic-plastic joints in statics and dynamics of beams with cracks. Marine Tech Trans 13 (2002) 53–67

    Google Scholar 

  5. Drewko, J.;Hien, T.D.: A stochastic eigenproblem of beams with cracked cross-sections. Theor Appl Mech 40(4) (2002): 986–1001

    Google Scholar 

  6. Drewko, J.;Sperski, M.: Vibration of multi-chamber shell structures with discontinuously variable cross-sections. Eng Trans 39 (1991) 163–180

    Google Scholar 

  7. Fox, R.L.;Kapoor, M.P.: Rate of change of eigenvalues and eigenvectors. AIAA 6(12) (1968) 2426–2429

    MATH  Google Scholar 

  8. Frank, P.M.: Introduction to system sensitivity theory. Academic, London 1978

    MATH  Google Scholar 

  9. Haftka, R.T.;Mroz, Z.: First- and second-order sensitivity analysis of linear and nonlinear structures. AIAA 24 (1986) 1187–1192

    MATH  MathSciNet  Google Scholar 

  10. Haug, G.J.; Choi, K.K.; Komkov, V.: Design sensitivity analysis of structural systems. Series in Math Sci Eng Academic, London 1986

  11. Hibbitt, Karlsson, and Sorensen, Inc.: ABAQUS. Standard users manual (2000) (License: TASK Academic Computer Centre)

    Google Scholar 

  12. Hien, T.D.;Kleiber, M.: Computational aspects in structural design sensitivity analysis for statics and dynamics. Comput Struct 33 (1989) 939–950

    Article  MATH  Google Scholar 

  13. Hildebrand, F.B.: Introduction to numerical analysis. McGraw-Hill, New York 1956

    MATH  Google Scholar 

  14. Krawczuk, M.: Dynamics of constructional elements with fatigue cracks. Rep. No. 441/1398/94 Inst Fluid Flow Machines PAS 1994 (Habilitation thesis—in Polish)

  15. MacNeal-Schwendler, Inc.: NASTRAN. Reference manual 1994 (License: TASK Academic Computer Centre)

  16. Nelson, R.B.: Simplified calculation of eigenvector derivatives. AIAA 14(9) (1976) 1201–1205

    Article  MATH  MathSciNet  Google Scholar 

  17. Nowacki, W.: Dynamics of structures. Arkady (1972) (in Polish)

  18. Sih, G.C.: Handbook of stress intensity factors. Lihigh University, Bethlehem 1973

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Maritine Technology, Technical University of Szczecin, Poland al. Piastow 41, 71-065, Szczecin, Poland

    J. Drewko

  2. Department of Civil Engineering, Technical University of Szczecin, Poland al. Piastow 50, 70-311, Szczecin, Poland

    T. D. Hien

Authors
  1. J. Drewko
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  2. T. D. Hien
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Correspondence to J. Drewko.

Additional information

The support of this work by the State Committee for Scientific Research (KBN) under Grant No. 4-050-0148/17-98-00 is gratefully acknowledged.

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Drewko, J., Hien, T.D. First-and second-order sensitivities of beams with respect to cross-sectional cracks. Arch. Appl. Mech. 74, 309–324 (2005). https://doi.org/10.1007/BF02637033

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  • DOI: https://doi.org/10.1007/BF02637033

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