zbMATH — the first resource for mathematics

Reliable analysis of transverse vibrations of Timoshenko-Mindlin beams with respect to uncertain shear correction factor. (English) Zbl 0971.74041
Summary: We investigate the dependence of minimal eigenfrequencies on the shear correction factor for homogeneous prismatic beams with simply supported or clamped ends. It is proved that the eigenfrequency is simple, and its derivative with respect to the shear correction factor is positive.

74H45 Vibrations in dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Full Text: DOI
[1] Bliss, G.A., Definitely self-adjoint boundary value problems for systems of ordinary differential equations, Trans. am. math. soc., 28, 561-584, (1926) · JFM 52.0453.13
[2] L. Collatz, Eigenwertaufgaben mit Technischen Anwendungen, 2. Auflage, Akad. Verlagsges. Geest und Portig, Leipzig, 1963
[3] Haug, E.; Choi, K.K.; Komkov, V., Design sensitivity analysis of structural systems, (1986), Academic Press Orlando · Zbl 0618.73106
[4] I. Hlaváček, Reliable solutions of elliptic boundary value problems with respect to uncertain data, in: Proceedings of the Second World Congress of Nonlinear Analysts, Nonlinear Anal., Theory, Methods and Appl. 30 (1997) 3879-3890
[5] Hlaváček, I., Reliable solution of an elasto-plastic reissner – mindlin beam for the Hencky’s model with uncertain yield function, Appl. math., 43, 223-237, (1998) · Zbl 1042.74533
[6] Litvinov, V.G., Optimization in elliptic boundary value problems with applications to mechanics, (1987), Nauka Moscow, (in Russian) · Zbl 0688.49003
[7] V.V. Nesterenko, A theory for transverse vibrations of the Timoshenko beam, J. Appl. Math. Mech. 57 (1993) 669-677; translation from Prikl. Mat. Mekh. 57 (1993) 83-91 · Zbl 0811.73035
[8] Rakowski, J., The interpretation of the shear locking in beam elements, Comput. struct., 37, 769-776, (1990)
[9] L. Trabucho, J. M. Viaño, Mathematical Modelling of Rods, in: P.G. Ciarlet, J. L. Lions (Eds.), Handbook of Numerical Analysis, vol. IV, Elsevier, Amsterdam, 1996, pp. 487-974
[10] Traill-Nash, R.W.; Collar, A.R., The effects of shear flexibility and rotatory inertia on the bending vibrations of beams, Quart. J. mech. appl. math., 6, 186-222, (1953) · Zbl 0050.40504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.