Summary
For the geometrically nonlinear first approximation theory of elastic shells three energy-consistent large rotation shell variants are constructed. The governing shell equations are derived as Euler-Lagrange equations of an associated variational principle of stationary total potential energy. The numerical applicability is considered for a highly nonlinear shell problem. To incorporate the presented theories into the frame of shell models published in the literature a comparative analysis is carried out for a large number of shell equations.
Übersicht
Im Rahmen der geometrisch-nichtlinearen ersten Schalenapproximation werden drei energiekonsistente Schalentheorien für große Rotationen hergeleitet. Die Schalengleichungen werden als Euler-Lagrange Gleichungen eines zugehörigen Variationsprinzips vom stationären Wert der potentiellen Gesamtenergie hergeleitet. Die numerische Anwendbarkeit wird anhand eines stark-nichtlinearen Schalenbeispiels nachgewiesen. Um die in dieser Arbeit hergeleiteten Theorien entsprechend einordnen zu können, wird für eine größere Zahl von in der Literatur angegebenen Schalengleichungen eine vergleichende Untersuchung mit numerischer Auswertung durchgeführt.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Shilkrut, D.: Solutions of some stability problems in the theory of geometrically non-linear shells. Israel J. Techn. 18 (1980) 76–83
Stumpf, H.: On the linear and nonlinear stability analysis in the theory of thin elastic shells. Ing. Arch. 51 (1981) 195–213
Krätzig, W. B.; Basar, Y.; Wittek, U.: Nonlinear behaviour and elastic stability of shells. Buckling of Shells, Proc. of a state-of-the-Art Colloquium. Berlin, Heidelberg, New York: Springer 1982
Stein, E.; Berg, A.; Wagner, W.: Different levels of non-linear shell theory in finite element stability analysis. Buckling of Shells, Proc. of a state-of-the-Art Colloquium. Berlin, Heidelberg, New York: Springer 1982
Harte, B.: Doubly curved triangular finite elements for the linear and geometrically nonlinear analysis of general shell structures (in German). KIB-TWM 82-10, Ruhr-Univ. Bochum 1982
Nolte, L.-P.: Contribution to the derivation and comparative investigation of geometrically nonlinear shell theories undergoing large rotations (in German). IfM-Mitt. 39, Ruhr-Univ. Bochum 1983
John, F.: Estimates for the derivatives of the stresses in a thin shell and interior shell equations. Comm. Pure Appl. Math. 18 (1965) 235–267
Koiter, W. T.: A consistent first approximation in the general theory of thin elastic shells. Proc. IUTAM Symp. Delft 1959. Amsterdam: North-Holland 1980, pp. 12–33
Koiter, W. T.: On the nonlinear theory of thin elastic shells. Proc. Kon. Ned. Ak. Wet., Ser B, 69 (1966) 1–54
Pietraszkiewicz, W.: Finite rotations in the non-linear theory of thin shells. In “Thin Shell Theory, New Trends and Applications”, Wien: Springer 1980, pp. 153–208
Pietraszkiewicz, W.; Szwabowicz, M. L.: Entirely Lagrangian nonlinear theory of thin shells. Arch. Mech. 33 (1981) 273–288
Pietraszkiewicz, W.: On consistent approximations in the geometrically non-linear theory of shells. IFM-Mitt. 26, Ruhr-Univ. Bochum 1981
Koiter, W. T.; Simmonds, J. G.: Foundations of shell theory. In “Theoretical and Applied Mechanics”, Proc. 13th. IUTAM Congr., Moscow 1972, Berlin, Heidelberg, New York: Springer 1973 pp. 150–175
Marguerre, K.: Zur Theorie der gekrümmten Platte großer Formänderungen. Proc. 5th Int. Congr. of Appl. Mech. Cambridge/Mass. 1938. New York: Wiley and Sons 1939, pp. 93–101
Donnell, L. H.: Stability of thin walled tubes under torsion. NACA TR 479, 1933
Vlasov, V. Z.: General theory of shells and its application in engineering. NASA TT F-99, 1964
Sanders, J. L.: Nonlinear theories of thin shells. Quart. Appl. Math. 21 (1963) 21–36
Schmidt, R.: Variational principles for geometrically nonlinear theories of shells undergoing moderate rotations (in German). Dissertation Ruhr-Univ. Bochum 1980
Pietraszkiewicz, W.: Lagrangian description and incremental formulation in the non-linear theory of thin shells. Int. J. Non-linear Mech. 19 (1984) 115–140
Schmidt, B.: On geometrically non-linear theories for thin elastic shells. Proc. EUROMECH Coll. “Flexible Shells, Theory and Applications”, München 1983
Makowski, J.; Nolte, L.-P.; Stumpf, H.: Finite in-plane deformations of flexible rods — Insight into nonlinear shell problems. IfM-Mitt. 46, Ruhr-Univ. Bochum 1985
Stumpf, H.: General concept of energy-consistent elastic shell theories with buckling and post-buckling analysis. Lecture XIV th ICTAM 1984, Lyngby, Denmark
Nolte, L.-P.; Stumpf, H.: Energy-consistent large rotation shell theories in Lagrangean description. Mech. Res. Comm. 10 (1983) 213–221
Pietraszkiewiez, W.: A simplest consistent version of the geometrically non-linear theory of elastic shells undergoing large/small rotations. Z. Angew. Math. Mech. 63 (1983) T 200–202
Stumpf, H.: On the non-linear buckling and postbuckling analysis of thin elastic shells. Int. J. Non-Linear Mech. 19 (1984) 195–215
Nolte, L.-P.: Stability equations of the general geometrically non-linear first approximation theory of thin elastic shells. Z. Angew. Math. Mech. 63 (1983) T 79–82
Basar, Y.: A geometrically nonlinear shell theory (in German). KIB-Berichte 38/39, Essen: Vulkan 1981
Shapovalov, L. A.: On a simplest variant of equations of geometrically non-linear theory of thin shells (in Russian). Mekh. Tv. Tela 3, No. 1 (1968) 56–62
Chuyko, A. N.: On nonlinear equations for the theory of thin shells. Soviet Appl. Mech. 19 (1983) 66–71
Yaghmai, S.: Incremental analysis of large deformations in mechanics of solids with applications to axisymmetric shells of revolution. Dissertation, University of California, Berkeley 1969
Varpasuo, P.: Incremental analysis of axisymmetric shallow shells with varying strain-displacement equations. Comp. Meth. Appl. Mech. Eng. 21 (1980) 153–169
Nolte, L.-P.; Diekkämper, B.; Makowski, J.: Two efficient FE-formulations in the analysis of highly nonlinear deformation of thin elastic beams. Z. Angew. Math. Mech. 65 (1985) 162–164
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nolte, L.P., Makowski, J. & Stumpf, H. On the derivation and comparative analysis of large rotation shell theories. Ing. arch 56, 145–160 (1986). https://doi.org/10.1007/BF00537243
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00537243