Abstract
This paper presents the treatment of elastically restrained ends for the axially, loaded beam-buckling problem for the central finite difference beam model, the microstructured beam model, and Eringen’s nonlocal continuous beam model. The equivalence between the central finite difference beam model and the microstructured beam model is established herein, and these equivalent systems are regarded as belonging to one class of discrete systems since they become indistinguishable. Also, the continualized form of the discrete system is obtained by adopting the continualization method that is based on an exponential displacement function. Three approaches are then proposed for matching the discrete system with Eringen’s nonlocal continuous system for the beam-buckling problem. The approaches depend on the assumption made on the constancy/or varying Eringen’s small length scale coefficient e 0 as well as which one of the discrete or continuum system is taken as the reference system.
Similar content being viewed by others
References
Eringen A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Challamel N., Lerbet J., Wang C.M., Zhang Z.: Analytical length scale calibration of nonlocal continuum from a microstructured buckling model. Z. Angew. Math. Mech. 94, 402–413 (2014)
Challamel, N., Zhang, Z., Wang, C.M.: Nonlocal equivalent continua for buckling and vibration analyses of microstructured beams. J. Nanomech. Micromech. (2013). doi:10.1061/(ASCE)NM.2153-5477.0000062
Wang C.M., Zhang Z., Challamel N., Duan W.H.: Calibration of Eringen’s small length scale coefficient for initially stressed vibrating nonlocal Euler beams based on microstructured beam model. J. Phys. D Appl. Phys. 46, 345501 (2013)
Zhang Z., Challamel N., Wang C.M.: Eringen’s small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model. J. Appl. Phys. 114, 114920 (2013)
Zhang, Z., Wang, C.M., Challamel, N.: Eringen’s length scale coefficient for buckling of nonlocal rectangular plates from microstructured beam-grid model. J. Eng. Mech. (2014) (in press)
Silverman I.K.: Discussion on the paper of “Salvadori M.G., Numerical computation of buckling loads by finite differences, Transactions of the ASCE, 116, 590–636, 1951”. T. ASCE 116, 625–626 (1951)
Hencky H.: Über die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Gelenkkette. Der Eisenbau 11, 437–452 (1921)
El Naschie M.S.: Stress, Stability and Chaos in Structural Engineering: An Energy Approach. McGraw-Hill, New York (1990)
Leckie F.A., Lindberg G.M.: The effect of lumped parameters on beam frequencies. Aeronaut. Q. 14, 224–240 (1963)
Wang C.M., Wang C.Y., Reddy J.N.: Exact Solutions for Buckling of Structural Members. CRC Press, Boca Raton (2005)
Zhang Z., Wang C.M., Challamel N., Elishakoff I.: Obtaining Eringen’s length scale coefficient for vibrating nonlocal beams via continualization method. J. Sound Vib. 333, 4977–4990 (2014)
Seide P.: Accuracy of some numerical methods for column buckling. J. Eng. Mech. ASCE 101, 549–560 (1975)
Elishakoff I., Santoro R.: Error in the finite difference based probabilistic dynamic analysis: analytical evaluation. J. Sound Vib. 281, 1195–1206 (2005)
Santoro R., Elishakoff I.: Accuracy of the finite difference method in stochastic setting. J. Sound Vib. 291, 275–284 (2006)
Zhang Y.Y., Wang C.M., Challamel N.: Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model. J. Eng. Mech. ASCE 136, 562–574 (2010)
Challamel N., Wang C.M., Elishakoff I.: Discrete systems behave as nonlocal structural elements: bending, buckling and vibration analysis. Eur. J. Mech. A-Solids 44, 125–135 (2014)
Challamel, N., Zhang, Z., Wang, C.M., Reddy, J.N., Wang, Q., Michelitsch, T., Collect, B.: On non-conservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach. Arch. Appl. Mech. (2014). doi:10.1007/s00419-014-0862-x
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, C.M., Gao, R.P., Zhang, H. et al. Treatment of elastically restrained ends for beam buckling in finite difference, microstructured and nonlocal beam models. Acta Mech 226, 419–436 (2015). https://doi.org/10.1007/s00707-014-1195-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-014-1195-0