Skip to main content
SpringerLink
Log in

On the theory of rigid/perfectly plastic plates under uniformly distributed loads

Zur Theorie starr/ideal-plastischer Platten unter Gleichlast

  • Contributed Papers
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Summary

This paper examines the theory of rigid/perfectly plastic plates satisfying the ‘square’ or ‘Johansen’ yield criterion. Attention is focussed on the statically determinate ‘corner regime’ when the plate is subject to uniform distributed loads. The governing equations are semilinear and hyperbolic. The geometry of the characteristic networks is studied and two families of analytic solutions are obtained, expressed in terms of Jacobi elliptic functions. A general numerical procedure for constructing the characteristic networks is developed and used to derive new incomplete solutions for square and rectangular plates.

Zusammenfassung

Untersucht werden starr/ideal-plastische Platten unter Verwendung der “quadratischen” (“Johansen-”)Fließbedingung. Besondere Beachtung gilt den statisch bestimmten Randzonen bei Beanspruchung der Platte durch Gleichlast. Die Grundgleichungen sind semilinear und hyperbolisch. Die Geometrie des Netzes der Charakteristiken wird untersucht. Zwei Familien von analytischen Lösungen, in Form von Jacobischen elliptischen Funktionen, werden erhalten. Eine allgemeine numerische Methode zur Bestimmung des Netzes der Charakteristiken wird entwickelt und zur Herleitung zweier unvollständiger Lösungen der quadratischen und der Rechteckplatte verwendet.

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. Hopkins, H. G., andW. Prager: The load carrying capacities of circular plates. J. Mech. Phys. Solids2, 1–13 (1953).

    Google Scholar 

  2. Prager, W.: Introduction to Plasticity. Addison-Wesley. 1959.

  3. Hodge, P. G.: Limit Analysis of Rotationally Symmetric Plates and Shells. Prentice-Hall. 1963.

  4. Wood, R. H.: Plastic and Elastic Design of Slabs and Plates. Thames and Hudson. 1961.

  5. Sawczuk, A., andT. Jaeger: Grenztragfähigkeitstheorie der Platten. Berlin-Göttingen-Heidelberg: Springer. 1963.

    Google Scholar 

  6. Nielsen, M. P.: Limit Analysis of Reinforced Concrete Slabs. Acta Polytechnica Scandinavica, No. 26 (1964).

  7. Massonnet, Ch.: Complete solutions describing the limit state of reinforced concrete slabs. Mag. Concr. Res.19, 13–32 (1967).

    Google Scholar 

  8. Hopkins, H. G.: On the plastic theory of plates. Proc. Roy. Soc. Ser. A241, 153–179 (1957).

    Google Scholar 

  9. Schumann, W.: On limit analysis of plates. Quart. Appl. Math.16, 61–71 (1958).

    Google Scholar 

  10. Sawczuk, A., andP. G. Hodge: Limit analysis and yield-line theory. J. Appl. Mech.35, 357–362 (1968).

    Google Scholar 

  11. Mansfield, E. H.: Studies in collapse analysis of rigid-plastic plates with a square yield diagram. Proc. Roy. Soc. Ser.A 241, 311–338 (1957).

    Google Scholar 

  12. Wood, R. H.: A partial failure of limit analysis for slabs, and the consequences for future research. Mag. Concr. Res.21, 79–90 (1969).

    Google Scholar 

  13. Collins, I. F.: On an analogy between plane strain and plate bending solutions in rigid/perfect plasticity theory. Int. J. Solids Structures7, 1057–1073 (1971).

    Google Scholar 

  14. Fox, E. N.: Limit analysis for plates: a simple loading problem involving a complex exact solution. Phil. Trans. Roy. Soc.272, 463–492 (1972).

    Google Scholar 

  15. Johnson, W.: Upper bounds to the load for the transverse bending of flat rigid-perfectly plastic plates. Int. J. Mech. Sci.11, 913–938 (1969).

    Google Scholar 

  16. Ericksen, J. L.: Tensor Fields, Principles of Classical Mechanics and Field Theory, Handbuch der Physik, Vol. III/1. Berlin-Göttingen-Heidelberg: Springer. 1960.

    Google Scholar 

  17. Shield, R. T.: Plate design for minimum weight. Quart. Appl. Maths.18, 131–144 (1960).

    Google Scholar 

  18. Hill, R.: The Mathematical Theory of Plasticity. Oxford University Press. 1950.

  19. Bowman, F.: Introduction to Elliptic Functions. English Universities Press. 1953.

  20. Weatherburn, C. E.: Differential Geometry, Vol. 1. Cambridge University Press. 1927.

  21. Green, A. P.: On the use of hodographs in problems of plane plastic strain. J. Mech. Phys. Solids2, 73–80 (1954).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Manchester Institute of Science and Technology, Sackville Street, Manchester, England

    I. F. Collins

Authors
  1. I. F. Collins
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

With 12 Figures

Rights and permissions

Reprints and permissions

About this article

Cite this article

Collins, I.F. On the theory of rigid/perfectly plastic plates under uniformly distributed loads. Acta Mechanica 18, 233–254 (1973). https://doi.org/10.1007/BF01178556

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01178556

Keywords

Search

Navigation