Stability of slender beams and frames resting on 2D elastic half-space. (English) Zbl 1293.74131
Summary: Making use of a mixed variational formulation based on the Green function of the substrate, which assumes as independent fields the structure displacements and the contact pressure, a simple and efficient finite element-boundary integral equation coupling method is derived and applied to the stability analysis of beams and frames resting on an elastic half-plane. Slender Euler-Bernoulli beams with different combinations of end constraints are considered. The examples illustrate the convergence to the existing exact solutions and provide new estimates of the buckling loads for different boundary conditions. Finally, nonlinear incremental analyses of rectangular pipes with compressed columns and free or pinned foundation ends are performed, showing that pipes stiffer than the soil may exhibit snap-through instability.
MSC:
| 74G60 | Bifurcation and buckling |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 74M15 | Contact in solid mechanics |
| 74S05 | Finite element methods applied to problems in solid mechanics |
Keywords:
buckling; Euler-Bernoulli beam; soil-structure interaction; frictionless contact; mixed finite elements; flamant solutionReferences:
| [1] | Wieghardt, K.: Über den Balken auf nachgiebiger Unterlage. ZAMM–Zeitschrift für AngewandteMathematik UndMechanik 2(3), 165–184 (1922) · JFM 48.0930.04 · doi:10.1002/zamm.19220020301 |
| [2] | Prager W.: Zur Theorie elastische gelagerter Konstruktionen. ZAMM–Zeitschrift für Angewandte Mathematik und Mechanik 7(5), 354–360 (1927) · JFM 53.0766.06 · doi:10.1002/zamm.19270070504 |
| [3] | Biot M.A.: Bending of an infinite beam on an elastic foundation. J. Appl. Mech. Trans. ASME 4, A1–A7 (1937) |
| [4] | Reissner M.E.: On the theory of beams resting on a yielding foundation. Proc. Natl. Acad. Sci. U.S.A. 23(6), 328–333 (1937) · JFM 63.0748.04 · doi:10.1073/pnas.23.6.328 |
| [5] | Gough G.S., Elam C.F., de Bruyne N.A.: The stabilisation of a thin sheet by a continuous supporting medium. J. R. Aeronaut. Soc. 44, 12–43 (1940) |
| [6] | Allen H.G.: Analysis and Design of Structural Sandwich Panels. Pergamon Press, Oxford (1969) |
| [7] | Ley, R.P., Lin, W., Mbanefo, U.: Facesheet wrinkling in sandwich structures. CR-1999-208994, NASA, Virginia (1999) |
| [8] | Davies J.M.: Lightweight Sandwich Construction. Blackwell Science, Oxford (2001) |
| [9] | Shield T.W., Kim K.S., Shield R.T.: The buckling of an elastic layer bonded to an elastic substrate in plane strain. J. Appl. Mech. TranS. ASME 61, 231–235 (1994) · doi:10.1115/1.2901434 |
| [10] | Volynskii A.L., Bazhenov S., Lebedeva O.V., Bakeev N.F.: Mechanical buckling instability of thin coatings deposited on soft polymer substrates. J. Mater. Sci. 35, 547–554 (2000) · doi:10.1023/A:1004707906821 |
| [11] | Timoshenko S.P., Gere J.M.: Theory of Elastic Stability. McGraw-Hill, New York (1961) |
| [12] | Hetenyi M.: Beam on Elastic Foundation. The University of Michigan Press, Ann Arbor (1946) |
| [13] | Goodier J.N., Hsu C.S.: Nonsinusoidal buckling modes of sandwich plates. J. Aeronaut. Sci. 21, 525–532 (1954) · Zbl 0057.16903 · doi:10.2514/8.3116 |
| [14] | Smith T.E.: Buckling of a beam on a Wieghardt-type elastic foundation. ZAMM–Zeitschrift für Angewandte Mathematik und Mechanik 49(11), 641–645 (1969) · Zbl 0207.23902 · doi:10.1002/zamm.19690491102 |
| [15] | Gallagher A.P.: Buckling of a beam under axial compression with elastic support. In: Scaife, B.K.P. (eds) Studies in Numerical Analysis, pp. 137–150. Academic Press, London (1974) · Zbl 0321.73037 |
| [16] | Bosakov S.V.: Variational approach to the solution of a contact problem for an elastic half-plane. Int. Appl. Mech. 30(7), 535–538 (1994) · doi:10.1007/BF00847249 |
| [17] | Tullini N., Tralli A.: Static analysis of Timoshenko beam resting on elastic half-plane based on coupling of locking-free elements and boundary integral. Comput. Mech. 45(2–3), 211–225 (2010) · Zbl 1271.74271 · doi:10.1007/s00466-009-0431-2 |
| [18] | Bazant Z.P., Cedolin L.: Stability of Structures. Oxford University Press, New York (1991) · Zbl 0455.73065 |
| [19] | Johnson K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985) · Zbl 0599.73108 |
| [20] | Reddy J.N.: An Introduction to the Finite Element Method, 3rd edn. McGraw Hill, Singapore (2006) |
| [21] | Vesic A.B.: Bending of beams resting on isotropic elastic solid. J. Eng. Mech. Div. ASCE 87(EM2), 35–53 (1961) |
| [22] | Stafford C.M., Harrison C., Beers K.L., Karim A., Amis E.J., VanLandingham M.R., Kim H.-C., Volksen W., Miller R.D., Simonyi E.E.: A buckling-based metrology for measuring the elastic moduli of polymeric thin films. Nat. Mater. 3(8), 545–550 (2004) · doi:10.1038/nmat1175 |
| [23] | Tullini, N., Tralli, A., Baraldi, D.: Buckling of Timoshenko beams in frictionless contact with an elastic half-plane. J. Eng. Mech. ASCE. doi: 10.1061/(ASCE)EM.1943-7889.0000529 |
| [24] | Simitses G.J.: An Introduction to Elastic Stability of Structures. Prentice-Hall, Englewood Cliffs (1976) |
| [25] | Batoz J.L., Dhatt G.: Incremental displacement algorithms for non-linear problems. Int. J. Numer. Methods Eng. 14(8), 1262–1267 (1979) · Zbl 0423.73061 · doi:10.1002/nme.1620140811 |
| [26] | Tullini N., Tralli A., Lanzoni L.: Interfacial shear stress analysis of bar and thin film bonded to 2D elastic substrate using a coupled FE-BIE method. Finite Elem. Anal. Des. 55, 42–51 (2012) · doi:10.1016/j.finel.2012.02.006 |
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