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Zur Lösung des Kontaktproblems elastischer Körper mit ausgedehnter Kontaktfläche durch quadratische Programmierung

To the solution of the contact problem of elastic bodies with extended contact areas by quadratic programming

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Zusammenfassung

Der mit einer elastischen oder starren Unterlage in Kontakt tretende Körper wird durch finite Elemente repräsentiert. Die totale potentielle Energie des Systems ist bei elastischem Materialverhalten und kleinen Verformungen eine quadratische Funktion in den Knotenvariablen und den Knotenpunktsordinaten der durch ein Polynom approximierten Kontaktdruckverteilung. Dazu ist nur ein Bereich auf der Oberfläche des Körpers festzulegen, innerhalb dessen die aktuelle Kontaktfläche liegen muß. Nach Ermittlung der Gleichgewichtsbedingungen und der entsprechenden Kontaktbedingung als Ungleichung sowie einer linearen Transformation der Knotenvariablen werden num sämtliche Beziehungen so formuliert, daß sich die Minimierung der totalen potentiellen Energie als ein Quadratisches Programm in den Knotenvariablen und den entsprechenden Ordinaten der Kontaktdruckverteilung darstellen läßt.

Die Lösung des quadratischen Programmes kann mit Standardprogrammen durchgeführt werden.

Abstract

An elastic body in contact with an elastic or rigid subgrade is represented by Finite Elements. The total potential energy of the system under consideration of linearly elastic material and small deformations is now a quadratic function of the nodal deformations and the nodal values of the contact pressure which is approximated by a polynomial. Only a part of the surface of the body must be proposed which includes the real contact surface. After evaluation of the equilibrium equations and the contact condition in an inequality and a linear transformation of the nodal variables, all relations are now so formulated, that minimization of total potential energy can be expressed as a Quadratic Program in the unknown nodal deformations and nodal values of contact pressure. Standard Programs can now be used for solution.

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Literatur

  1. Timoshenko, S. P.: Theory of Elasticity, 2. Aufl., S. 372 ff. New York: McGraw-Hill 1951.

    Google Scholar 

  2. Galin, L. A.: Contact Problems in the Theory of Elasticity (Übersetzung aus dem Russischen von Moss, H.), 1. Aufl. North Carolina State College 1961.

  3. Dundurs, J., Stippes, M.: Role of Elastic Constants in Certain Contact Problems. J. Appl. Mech.37, 4, 965–970 (1970).

    Google Scholar 

  4. Parlas, S. C., Michalopoulos, C. D.: Axisymmetric Contact Problem for an Elastic Half-Space with a Cylindrical Hole. Int. J. Engng. Sci.10, 8, 699–707 (1972).

    Article  Google Scholar 

  5. Ollerton, E.: Elastic Stresses between Curved Bodies in Rolling Contact. Strain7, 2, 80–87 (1971).

    Google Scholar 

  6. Kuznetsov, V. N.: The Elastic Stress Field in the Plane Contact Problem. Strength of Materials3, 2, 221–225 (1971).

    Article  Google Scholar 

  7. Vorovich, I. I., Penin, O. M.: Contact Problems for an Infinite Strip of Varying Height (in Russian). Izvestiya Akademii Nauk SSR, Mekhanika Tverdogo Tela5, 112–121 (1971).

    Google Scholar 

  8. Schwartz, J., Harper, E. Y.: On the Relative Approach of Two-Dimensional Elastic Bodies in Contact. Int. J. Solids Structures7, 1613–1626 (1971).

    Article  Google Scholar 

  9. Kalker, J. J.: On Elastic Line Contact. J. Appl. Mech.39, 3, 1125–1132 (1972).

    Google Scholar 

  10. Schindler, K.: Der Träger auf Elastischer Bettung unter Ausschluß von Zugkräften zwischen Träger und Unterlage. Österreichische Bauzeitschrift11, 3, 47–53 (1956).

    Google Scholar 

  11. Tsai, N., Westmann, R. A.: Beam on Tensionless Foundation. J. Engng. Mech. Div.93, 5, 1–12 (1967).

    Google Scholar 

  12. Weitsman, Y.: On Foundations that React in Compression Only. J. Appl. Mech.37, 4, 1019–1030 (1970).

    Google Scholar 

  13. Weitsman, Y.: Onset of Separation between a Beam and a Tensionless Elastic Foundation under a Moving Load. Int. J. Mech. Sci.13, 8, 707–711 (1971).

    Article  Google Scholar 

  14. Weitsman, Y.: A Tensionless Contact between a Beam and a Elastic Halfspace. Int. J. Engng. Sci.10, 1, 73–81 (1972).

    Article  Google Scholar 

  15. Pu, S. L., Hussain, M. A., Anderson, G.: Lifting of a Plate from the Foundation Due to Axisymmetric Pressure. Developments in Mechanics, Vol. 5 (Proc. of the 11th Midwestern Mechanics Conf.), S. 577–590 (1969).

    Google Scholar 

  16. Weitsman, Y.: On the Unbonded Contact between Plates and a Elastic Half Space. J. Appl. Mech.36, 2, 199–202 (1969).

    Google Scholar 

  17. Hussain, M. A., Pu, S. L.: A Variational Principle for Singular Integral Equations with Bounded Solutions. Int. J. Engng. Sci.11, 7, 767–781 (1973).

    Article  Google Scholar 

  18. Keer, L. M., Chantaramungkorn, K.: Loss of Contact between an Elastic Layer and Half Space. J. Elasticity2, 3, 191–197 (1972).

    Google Scholar 

  19. Hussain, M. A., Pu, S. L.: Slip Phenomenon for a Circular Inclusion. J. Appl. Mech.38, 6, 627–633 (1971).

    Google Scholar 

  20. Wilson, H. B., Jr., Hill, J. L.: Non-Hertzian Contact Stress in a Smoothly Cradled Heavy Cylinder. J. Elasticity2, 2, 143–151 (1972).

    Article  Google Scholar 

  21. Brandes, K.: Die Lagerung des Kreiszylinderrohres auf einem starren Linienlager, Beitrag zur praktischen Berechnung der Kontaktkräfte. Stahlbau40, 10, 298–310 (1971).

    Google Scholar 

  22. Grigoliuk, E. I., Tolkachev, V. M.: Contact Problem for Semi-Infinite Cylinder Shell. PMM J. Appl. Math. Mech.35, 5, 785–794 (1971).

    Article  Google Scholar 

  23. Kulkarni, S. V., Frederick, D.: The Contact Problem of Two Coaxial Cylindrical Shells. Int. J. Mech. Sci.15, 5, 367–379 (1973).

    Article  Google Scholar 

  24. Updike, D. P., Kalnins, A.: Contact Pressure between an Elastic Shell and a Rigid Plate. J. Appl. Mech.39, 4, 1110–1114 (1972).

    Google Scholar 

  25. Duncan, J. M., Clough, G. W.: Finite Element Analysis of Port Allen Lock. J. Soil Mech. Found. Div.97, SM8, 1053–1068 (1971).

    Google Scholar 

  26. Chan, S. K., Tuba, I. S.: A Finite Element Method for Contact Problems of Solid Bodies —Part I. Theory and Validation. Int. J. Mech. Sci.13, 7, 615–625 (1971);—Chan, S. K., Tuba, I. S.: A Finite Element Method for Contact Problems of Solid Bodies — Part II. Application to Turbine Blade Fastenings. Int. J. Mech. Sci.13, 7, 627–639 (1971).

    Article  Google Scholar 

  27. Gould, H. H., Mikic, B. B.: Areas of Contact and Pressure Distribution in Bolted Joints. J. Eng. Ind.94, 3, 864–870 (1972).

    Google Scholar 

  28. Wright, G. P., O'Connor, J. J.: Finite Element Analysis of Alternating Axial Loading of an Elastic Plate Pressed between Two Rectangular Blocks with Finite Friction. Int. J. Engng. Sci.9, 6, 555–570 (1971).

    Article  Google Scholar 

  29. Svec, O. J., Gladwell, G. M. L.: A Triangular Plate Bending Element for Contact Problems. Int. J. Solids Structures9, 3, 435–446 (1973).

    Article  Google Scholar 

  30. Svec, O. J., Gladwell, G. M. L.: An Explicit Boussinesq Solution for a Polynomial Distribution of Pressure over a Triangular Region. J. Elasticity1, 2, 167–170 (1971).

    Article  Google Scholar 

  31. Svec, O. J.: The Unbonded Contact Problem of a Plate on the Elastic Half Space. Comp. Meth. Appl. Mech. Engng.3, 105–113 (1974).

    Article  Google Scholar 

  32. Wilson, E. A., Parsons, B.: Finite Element Analysis of Elastic Contact Problems Using Differential Displacements. Int. J. Num. Meth. Engng.2, 3, 387–395 (1970).

    Article  Google Scholar 

  33. Gangal, M. D.: Direct Finite Element Analysis of Elastic Contact Problems. Int. J. Num. Meth. Engng.5, 1, 145–147 (1972).

    Article  Google Scholar 

  34. Hadley, G.: Nichtlineare und Dynamische Programmierung, 1. Aufl. Würzburg-Wien: Physica Verlag 1969.

    Google Scholar 

  35. Shield, R. T.: Optimum Design of Structures through Variational Principles (Lecture Notes in Physics. Vol. 21). S. 13–37. Berlin-Heidelberg-New York: Springer 1973.

    Google Scholar 

  36. Schiras, A. A.: Metody lineinogo programmirovaniya pri raschete uprugoplasticheskikh sistem (“Methoden der linearen Programmierung für die Berechnung elastoplastischer Systeme”), 1. Aufl. Leningrad: Stroiizdat 1969.

    Google Scholar 

  37. Anderheggen, E., Knopfel, H.: Finite Element Limit Analysis Using Linear Programming. Int. J. Solids Structures8, 12, 1413–1431 (1972).

    Article  Google Scholar 

  38. Fox, R. L., Stanton, E. L.: Developments in Structural Analysis by Direct Energy Minimization. AIAA J.6, 6, 1036–1042 (1968).

    Google Scholar 

  39. Gisvold, K. M., Moe, J.: Buckling Analysis by Means of Nonlinear Programming. Int. J. Num. Math. Engng.2, 3, 353–361 (1970).

    Article  Google Scholar 

  40. Szabo, B. A., Tsai, C.-T.: The Quadratic Programming Approach to the Finite Element Method. Int. J. Num. Math. Engng.5, 3, 375–381 (1973).

    Article  Google Scholar 

  41. Nitsiotas, G.: Die Berechnung statisch unbestimmter Tragwerke mit einseitigen Bindungen. Ingenieur-Archiv41, 1, 46–60 (1971).

    Article  Google Scholar 

  42. Zarghamee, M. S.: Analysis of Structures by Quadratic Programming. J. Engng. Mech. Div.97, 5, 1495–1502 (1971).

    Google Scholar 

  43. Sayegh, A. F.: Elastic Analysis with Indeterminate Boundary Conditions. J. Engng. Mech. Div.100, 1, 49–62 (1974).

    Google Scholar 

  44. Conry, T. F., Seireg, A.: A Mathematical Programming Method for Design of Elastic Bodies in Contact. J. Appl. Mech.38, 2, 387–392 (1971).

    Google Scholar 

  45. Kalker, J. J., Van Randen, Y.: A Minimum Principle for Frictionless Contact with Application to Non-Hertzian Half-Space Contact Problems. J. Engng. Math.6, 2, 193–206 (1972).

    Article  Google Scholar 

  46. Beale, E. M. L.: On Minimizing a Convex Function Subject to Linear Inequalities. J. Roy. Stat. Soc.17b, 173–184 (1955). Lemke, C. E.: On Complementary Pivot Theory, in: Mathematics of the Decision Science (Dantzig, G. B., Veinott A. F., Hrsg.). American Mathematical Society. 1967. Land, A. H., Powell, S.: Fortran-Codes for Mathematical Programming. London-New York: J. Wiley 1973.

    Google Scholar 

  47. Baranenko, V. A., Pochtman, Y. M.: Analysis of Deformation State of Elastic Plates and Membranes, Limited by Restrictions, by Dynamic Programming. Izvestiya Akademii Nauk Armyanskoi SSR, Mekhanika23, 4, 44–49 (1970).

    Google Scholar 

  48. Waters, E. O.: Derivation of Code Formulas for Part B Flanges, Welding Research Council Bulletin166, 27–37 (1971).

    Google Scholar 

  49. Künzi, H. P., Tzschach, H. G., Zehnder, C.-A.: Numerische Methoden der mathematischen Optimierung. 1. Aufl., S. 63ff. Stuttgart: B. G. Teubner 1967.

    Google Scholar 

  50. Zienkiewicz, O. C.: The Finite Element Method in Engineering Science, 2. Aufl. London: McGraw-Hill 1971.

    Google Scholar 

  51. Buck, K., Scharpf, D. W., Stein, E., Wunderlich, W.: Finite Elemente in der Statik, 1. Aufl. Berlin-München-Düsseldorf: W. Ernst & Sohn 1973.

    Google Scholar 

  52. Zienkiewicz, O. C.: Introductory Lectures on the Finite Element Method. (CISM No. 130.) Wien-New York: Springer 1972.

    Google Scholar 

  53. Fenves, S. J.: A Structural Analysis by Networks, Matrices and Computers. J. Struct. Div.92, 199–221 (1966).

    Google Scholar 

  54. Orlov, G., Saxenhofer, H.: Balken auf elastischer Unterlage, S. 42ff. Zürich: Leemann 1969.

    Google Scholar 

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    F. D. Fischer

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Fischer, F.D. Zur Lösung des Kontaktproblems elastischer Körper mit ausgedehnter Kontaktfläche durch quadratische Programmierung. Computing 13, 353–384 (1974). https://doi.org/10.1007/BF02241725

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