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A finite element lumped mass scheme for solving eigenvalue problems of circular arches. (English) Zbl 0438.73065


MSC:

74S05 Finite element methods applied to problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs

Keywords:

circular arches
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References:

[1] Archer, R.R.: Small vibrations of thin incomplete circular rings. Internat. J. Mechanical Sciences1, 45-56 (1960) · doi:10.1016/0020-7403(60)90029-1
[2] Ciarlet, P.G., Schultz, M.H., Varga, R.S.: Numerical methods of high-order accuracy for nonlinear boundary value problems III. Eigenvalue problems. Numer. Math.12, 120-133 (1968) · Zbl 0181.18303 · doi:10.1007/BF02173406
[3] Ciarlet, P.G.: The finite element method for elliptic problems, Amsterdam: North-Holland 1978 · Zbl 0383.65058
[4] Collatz, L.: The numerical treatment of differential equations. Berlin Heidelberg New York: Springer 1960 · Zbl 0086.32601
[5] Dawe, D.J.: Numerical studies using circular arch finite elements. Computers & Structures4, 729-740 (1974) · doi:10.1016/0045-7949(74)90041-8
[6] Den Hartog, J.P.: The lowest natural frequency of circular arcs. Philosophical Magazine5 (Series 7), 400-408 (1928) · JFM 54.0856.06
[7] Gellert, M., Laursen, M.E.: Formulation and convergence of a mixed finite element method applied to elastic arches of arbitrary geometry and loading. Comput. Methods Appl. Mech. Engrg.7, 285-302 (1976) · Zbl 0339.73042 · doi:10.1016/0045-7825(76)90064-5
[8] Ishihara, K.: A mixed finite element method for the biharmonic eigenvalue problems of plate bending. Publ. Res. Inst. Math. Sci.14, 399-414 (1978) · Zbl 0389.73075 · doi:10.2977/prims/1195189071
[9] Ishihara, K.: On curved finite element and straight beam element approximations for vibration problems of circular arch structures. J. Math. Kyoto Univ.20, 753-782 (1980) · Zbl 0457.73053
[10] Kikuchi, F.: On the validity of the finite element analysis of circular arches represented by an assemblage of beam elements. Comput. Methods Appl. Mech. Engrg.5, 253-276 (1975) · Zbl 0301.73023 · doi:10.1016/0045-7825(75)90001-8
[11] Krieg, R.D., Key, S.W.: Transient shell response by numerical time integration. Internat. J. Numer. Methods Engrg.7, 273-286 (1973) · doi:10.1002/nme.1620070305
[12] Moan, T.: A note on the convergence of finite element approximations for problems formulated in curvilinear coordinate systems. Comput. Methods Appl. Mech. Engrg.3, 17-30 (1971)
[13] Sabir, A.B., Ashwell, D.G.: A comparison of curved beam finite elements when used in vibration problems. J. Sound Vibration18, 555-563 (1971) · doi:10.1016/0022-460X(71)90106-4
[14] Schultz, M.H.: Spline analysis. London: Prentice-Hall 1973 · Zbl 0333.41009
[15] Strang, G., Fix, G.: An analysis of the finite element method. New York: Prentice-Hall 1973 · Zbl 0356.65096
[16] Surana, K.S.: Lumped mass matrices with non-zero inertia for general shell and axisymmetric shell elements. Internat. J. Numer. Methods Engrg.12, 1635-1650 (1978) · Zbl 0385.73079 · doi:10.1002/nme.1620121102
[17] Timoshenko, S., Woinowsky-Krieger, S., Theory of plates and shells. New York: McGraw-Hill 1959 · Zbl 0114.40801
[18] Volterra, E., Morell, J.D.: A note on the lowest natural frequency of elastic arcs. Trans. ASME Ser. E. J. Appl. Mech.27, 744-746 (1960) · Zbl 0100.21304
[19] Wilkinson, J.H.: The algebraic eigenvalue problem. Oxford: Oxford University Press 1965 · Zbl 0258.65037
[20] Yosida, K.: Functional analysis. Berlin Heidelberg New York: Springer 1968 · Zbl 0152.32102
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