Skip to main content
SpringerLink
Log in

Compatibility Condition in Theory of Solid Mechanics (Elasticity, Structures, and Design Optimization)

  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

The strain formulation in elasticity and the compatibility condition in structural mechanics have neither been understood nor have they been utilized. This shortcoming prevented the formulation of a direct method to calculate stress and strain, which are currently obtained indirectly by differentiating the displacement. We have researched and understood the compatibility condition for linear problems in elasticity and in finite element structural analysis. This has lead to the completion of the “method of force” with stress (or stress resultant) as the primary unknown. The method in elasticity is referred to as the completed Beltrami-Michell formulation (CBMF), and it is the integrated force method (IFM) in the finite element analysis. The dual integrated force method (IFMD) with displacement as the primary unknown had been formulated. Both the IFM and IFMD produce identical responses. The IFMD can utilize the equation solver of the traditional stiffness method. The variational derivation of the CBMF produced the existing sets of elasticity equations along with the new boundary compatibility conditions, which were missed since the time of Saint-Venant, who formulated the field equations about 1860. The CBMF, which can be used to solve stress, displacement, and mixed boundary value problems, has eliminated the restriction of the classical method that was applicable only to stress boundary value problem. The IFM in structures produced high-fidelity response even with a modest finite element model. Because structural design is stress driven, the IFM has influenced it considerably. A fully utilized design method for strength and stiffness limitation was developed via the IFM analysis tool. The method has identified the singularity condition in structural optimization and furnished a strategy that alleviated the limitation and reduced substantially the computation time to reach the optimum solution. The CBMF and IFM tensorial approaches are robust formulations because both methods simultaneously emphasize the equilibrium equation and the compatibility condition. The vectorial displacement method emphasized the equilibrium, while the compatibility condition became the basis of the scalar stress-function approach. The tensorial approach can be transformed to obtain the vector and the scalar methods, but the reverse course cannot be followed. The tensorial approach outperformed other methods as expected. This paper introduces the new concepts in elasticity, in finite element analysis, and in design optimization with numerical illustrations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Timoshenko SP (1983) History of strength of materials. Dover, New York

    Google Scholar 

  2. Ferrers NM (ed) (1871) Mathematical papers of the late George Green. Macmillan, London

    Google Scholar 

  3. Patnaik SN (1986) The variational energy formulation for the integrated force method. AIAA J 24(1):129–137

    MATH  MathSciNet  Google Scholar 

  4. Przemieniecki JS (1968) Theory of matrix structural analysis. McGraw-Hill, New York

    MATH  Google Scholar 

  5. Patnaik SN, Hopkins DA (2004) Strength of materials, a unified theory for the 21st century. Elsevier, Amsterdam

    Google Scholar 

  6. Patnaik SN, Hopkins DA (2005) Completed Beltrami Michell formulation in polar coordinates. NASA/TM-2005-213634

  7. Sokolnikoff IS (1956) Mathematical theory of elasticity, 2nd edn. McGraw-Hill, New York

    MATH  Google Scholar 

  8. Patnaik SN, et al. (1996) Completed Beltrami-Michell formulation for analyzing mixed boundary value problems in elasticity. AIAA J 34(1):143–148

    MATH  Google Scholar 

  9. Patnaik SN, Berke L, Gallagher RH (1991) Integrated force method versus displacement method for finite-element analysis. Comput Struct 38(4):377–407

    Article  MATH  Google Scholar 

  10. Patnaik SN (1986) The integrated force method versus the standard force method. Comput Struct 22(2):151–163

    Article  MATH  Google Scholar 

  11. Patnaik SN et al. (1992) Improved accuracy for finite-element structural-analysis via an integrated force method. Comput Struct 45(3):521–542

    Article  Google Scholar 

  12. Dayaratnam P, Patnaik S (1969) Feasibility of full stress design. AIAA J 7:773–774

    Google Scholar 

  13. Patnaik S, Dayaratnam P (1970) Behaviour and design of pin connected structures. Int J Numer Methods Eng 2(4):579–595

    Article  Google Scholar 

  14. Love AEH (1927) A treatise on the mathematical theory of elasticity. University Press, Cambridge

    MATH  Google Scholar 

  15. Ritz W (1909) Uber eine neue Methode zur Losung gewisser Variationsprobleme der mathematischen Physik. J Reine Angew Math 135:1–61

    Google Scholar 

  16. Timoshenko SP (1940) Theory of plates and shells. McGraw-Hill, New York

    MATH  Google Scholar 

  17. Park KC, Felippa CA (2000) A variational principle for the formulation of partitioned structural systems. Int J Numer Methods Eng 47:395–418

    Article  MATH  MathSciNet  Google Scholar 

  18. Denke PH (1959). A general digital computer analysis of statically indeterminate structures. NASA TN D-1666

  19. Patnaik SN, Yadagiri S (1982) Frequency-analysis of structures by integrated force. Method J Sound Vib 83(1):93–109

    Article  Google Scholar 

  20. Patnaik SN, Berke L, Gallagher RH (1991) Compatibility conditions of structural mechanics for finite-element analysis. AIAA J 29(5):820–829

    Article  Google Scholar 

  21. Hopkins D, et al. (2002) Fidelity of the integrated force method solution. Int J Numer Methods Eng 55(11):1367–1371

    Article  MATH  Google Scholar 

  22. Patnaik SN, Coroneos RM, Hopkins DA (1997) Dynamic animation of stress modes via the integrated force method of structural analysis. Int J Numer Methods Eng 40(12):2151–2169

    Article  MATH  Google Scholar 

  23. Storaasli OO, Baddourah M, Nguyen D (1995) GPS: general purpose solver. NASA Langley Software

  24. HARWELL subroutine library specifications (1990) Harwell Laboratory, Oxfordshire, OX11 ORA, England

  25. Batoz J-L, Dhatt G (1990) Modélisation des structures par éléments finis. Solides élastiques, vol 1. Hermes, Paris

    Google Scholar 

  26. Cilley FH (1900) The exact design of statically indeterminate frameworks. An exposition of its possibility, but futility. Trans Am Soc Civil Eng 42:353–407

    Google Scholar 

  27. Reinschmidt KF, Cornell CA, Brotchie JF (1966) Iterative design and structural optimization, American society of civil engineering. J Struct Div 92(281):281–318

    Google Scholar 

  28. Gallagher RH (1973) Fully stressed design. In: Gallagher RH, Zienkiewicz OC (eds) Optimum structural design. Wiley, London, pp 19–32

    Google Scholar 

  29. Patnaik SN (1986) Behavior of trusses with stress and displacement constraints. Int J Comput Struct 22:619–623

    Article  Google Scholar 

  30. Patnaik SN, Guptill JD, Berke L (1993) Singularity in structural optimization. Int J Numer Methods Eng 36:931–944

    Article  MATH  Google Scholar 

  31. Patnaik SN, Gendy AS, Hopkins DA (1998) Modified fully utilized method (MFUD) for stress and displacement constraints. Int J Numer Methods Eng 41:1171–1194

    Article  MATH  Google Scholar 

  32. Patnaik SN, Pai SS, Hopkins DA (2006). Precision of sensitivity in the design optimization of indeterminate structures. NASA/TP-2006-213818

  33. Patnaik SN, Gallagher RH (1988) Gradients of behavior constraints and reanalysis via the integrated force method. Int J Numer Methods Eng 23:2205–2212

    Article  Google Scholar 

  34. Patnaik SN, Hopkins DA (2001). Stress formulation in three-dimensional elasticity. NASA/TP-2001-210515

  35. Washizu K (1958) A note on the conditions of compatibility. J Math Phys 36:306–312

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ohio Aerospace Institute, 22800 Cedar Point Road, Brookpark, OH, 44142, USA

    Surya N. Patnaik

  2. NASA Glenn Research Center, 21000 Brookpark Road, Cleveland, OH, 44135, USA

    Shantaram S. Pai & Dale A. Hopkins

Authors
  1. Surya N. Patnaik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shantaram S. Pai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dale A. Hopkins
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Surya N. Patnaik.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Patnaik, S.N., Pai, S.S. & Hopkins, D.A. Compatibility Condition in Theory of Solid Mechanics (Elasticity, Structures, and Design Optimization). Arch Computat Methods Eng 14, 431–457 (2007). https://doi.org/10.1007/s11831-007-9011-9

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-007-9011-9

Keywords

Search

Navigation