Abstract
In this paper, a size-dependent Timoshenko beam is developed on the basis of the couple stress theory. The couple stress theory is a non-classic continuum theory capable of capturing the small-scale size effects on the mechanical behavior of structures, while the classical continuum theory is unable to predict the mechanical behavior accurately when the characteristic size of structures is close to the material length scale parameter. The governing differential equations of motion are derived for the couple-stress Timoshenko beam using the principles of linear and angular momentum. Then, the general form of boundary conditions and generally valid closed-form analytical solutions are obtained for the axial deformation, bending deflection, and the rotation angle of cross sections in the static cases. As an example, the closed-form analytical results are obtained for the response of a cantilever beam subjected to a static loading with a concentrated force at its free end. The results indicate that modeling on the basis of the couple stress theory causes more stiffness than modeling by the classical beam theory. In addition, the results indicate that the differences between the results of the proposed model and those based on the classical Euler–Bernoulli and classical Timoshenko beam theories are significant when the beam thickness is comparable to its material length scale parameter.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fleck N.A., Muller G.M., Ashby M.F., Hutchinson J.W.: Strain gradient plasticity: theory and experiment. J. Acta Metall. Mater. 42(2), 475–487 (1994)
Stolken J.S., Evans A.G.: Microbend test method for measuring the plasticity length scale. J. Acta Mater. 46(14), 5109–5115 (1998)
Chong A.C.M., Lam D.C.C.: Strain gradient plasticity effect in indentation hardness of polymers. J. Mater. Res. 14(10), 4103–4110 (1999)
McFarland A.W., Colton J.S.: Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 15(5), 1060–1067 (2005)
Kong S., Zhou S., Nie Z., Wang K.: The size-dependent natural frequency of Bernoulli—Euler micro-beams. Int. J. Eng. Sci. 46, 427–437 (2008)
Mindlin R.D.: Influence of couple stress on stress concentration. Exp. Mech. 3, 1–7 (1963)
Eringen A.C.: Theory of micropolar elasticity. In: Leibowitz, H. (eds) Fracture, vol. 2, pp. 621–629. Academic Press, New York (1968)
Yang F., Chong A.C.M., Lam D.C.C., Tong P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002)
Ariaei, A., Ziaei-Rad, S., Ghayour, M.: Transverse vibration of a multiple-Timoshenko beam system with intermediate elastic connections due to a moving load. Arch. Appl. Mech. (2009). doi:10.1007/s00419-010-0410-2
Ozdemir Ozgumus O., Kaya M.O.: Flapwise bending vibration analysis of a rotating double-tapered Timoshenko beam. Arch. Appl. Mech. 78, 379–392 (2008). doi:10.1007/s00419-007-0158
Lim C.W., Wang C.M., Kitipornchai S.: Timoshenko curved beam bending of Euler-Bernoulli solutions in terms. Arch. Appl. Mech. 67, 179–190 (1997)
Müller M., Pao Y.-H., Hauger W.: A dynamic model for a Timoshenko beam in an elastic-plastic state. Arch. Appl. Mech. 63, 301–312 (1993)
Anthoine A.: Effect of couple-stresses on the elastic bending of beams. Int. J. Solids Struct. 37, 1003–1018 (2000)
Park S.K., Gao X.L.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16(11), 2355–2359 (2006)
Ramezani S., Naghdabadi R., Sohrabpour S.: Analysis of micropolar elastic beams. Eur. J. Mech. A/Solids 28, 202–208 (2009)
Ma H.M., Gao X.-L., Reddy J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids V 56, 3379–3391 (2008)
Ramezani S., Naghdabadi R., Sohrabpour S.: Non-linear finite element implementation of micropolar hypo-elastic materials. J. Comput. Methods Appl. Mech. Eng. 197, 4149–4159 (2008)
McElhaney K.W., Valssak J.J., Nix W.D.: Determination of indenter tip geometry and indentation contact area for depth sensing indentation experiments. J. Mater. Res. 13, 1300–1306 (1998)
Nix W.D., Gao H.: Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411–425 (1998)
Maranganti R., Sharma P.: A novel atomistic approach to determine strain-gradient elasticity constants: Tabulation and comparison for various metals, semiconductors, silica, polymers and the (Ir) relevance for nanotechnologies. J. Mech. Phys. Solids 55, 1823–1852 (2007)
Lam D.C.C, Yang F., Chong A.C.M., Wang J., Tong P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Asghari, M., Kahrobaiyan, M.H., Rahaeifard, M. et al. Investigation of the size effects in Timoshenko beams based on the couple stress theory. Arch Appl Mech 81, 863–874 (2011). https://doi.org/10.1007/s00419-010-0452-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-010-0452-5