Free vibration and buckling analysis of functionally graded deep beam-columns on two-parameter elastic foundations using the differential quadrature method
Restricted accessResearch articleFirst published online June, 2009
Free vibration and buckling analysis of functionally graded deep beam-columns on two-parameter elastic foundations using the differential quadrature method
In this research, a differential quadrature method is applied for free vibration and buckling analysis of deep beam-columns composed of functionally graded materials on two-parameter elastic foundations. Derivation of equations is based on the unconstrained higher-order shear deformation theory taking into account the complete effects of shear deformation, depth change, and rotary inertia. It is assumed that the effective mechanical properties of functionally graded (FG) beam-columns are temperature dependent and vary continuously throughout the thickness direction according to volume fraction of the constituents defined by power-law function. The accuracy, convergence, and flexibility of the differential quadrature technique for simply supported FG deep beam-columns with complicated governing differential equations and boundary conditions are examined and verified with the known data in the literature.
FukuiY., ‘Fundamental investigation of functionally gradient material manufacturing system using centrifugal force’ Int. J. Jpn. Soc. Mech. Eng. III34 (1991): 144–148.
4.
TimoshenkoS. P., ‘On the correction for shear of the differential equation for transverse vibrations of prismatic bars’ Philos. Mag.41 (1921): 744–746.
5.
TimoshenkoS. P., ‘On the transverse vibrations of bars of uniform cross-section’ Philos. Mag.43 (1922): 125–131.
6.
ReissnerE., ‘The effect of transverse shear deformation on the bending of elastic plates’ J. Appl. Mech.12 (1945): 69–77.
7.
HenckyH., ‘Uber die berucksichtigung der schubverzerrung in ebenen platen’ Ing.-Arch.16 (1947): 72–76.
8.
MindlinR. D., ‘Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates’ J. Appl. Mech.18 (1951): 31–38.
9.
WangC. M.LimG. T.ReddyL. N.LeeK. H., ‘Relationships between bending solutions of Reissner and Mindlin plate theories’ Eng. Struct.23 (2001): 838–849.
10.
LevinsonM., ‘A new rectangular beam theory’ J. Sound Vibr.74 (1981): 81–87.
11.
ReddyJ. N., Energy and variational methods in applied mechanics (New York: John Wiley and Sons1984).
12.
BickfordW. B., ‘A consistent higher-order beam theory’ J. Appl. Mech.11 (1982): 137–150.
13.
HeyligerP. R.ReddyJ. N., ‘A higher-order beam finite element for bending and vibration problems’ J. Sound Vibr.126 (1988): 309–326.
14.
KhdeirA. A.ReddyJ. N., ‘Free vibration of cross-ply laminated beams with arbitrary boundary conditions’ Int. J. Eng. Struct.3212 (1980): 1971–1994.
15.
HannaN. F.LeissaA. W., ‘A higher-order shear deformation theory for the vibration of thick plates’ J. Sound Vibr.1704 (1994): 545–555.
16.
TesslerA.SaetherE.TsuiT., ‘Vibration of thick laminated composite plates’ J. Sound Vibr.1793 (1995): 475–498.
MatsunagaH., ‘Buckling instabilities of thick elastic plates subjected to in-plane stresses’ Comput. Struct.621 (1997): 205–214.
19.
MatsunagaH., ‘Free vibration and stability of laminated composite circular arches subjected to initial axial stress’ J. Sound Vibr.271 (2004): 651–670.
20.
RohwerK.RolfesR.SparrH., ‘Higher-order theories for thermal stresses in layered plates’ Int. J. Solids Struct.38 (2001): 3673–3687.
21.
BaoG.WangL., ‘Multiple cracking in functionally graded ceramic/metal coating’ Int. J. Solids Struct.32 (1995): 2853–2871.
22.
MatsunagaH., ‘Vibration and buckling of deep beam-columns on two-parameter elastic foundations’ J. Sound Vibr.2282 (1999): 359–376.
23.
ReddyJ. N., Theory and analysis of elastic plates (Philadelphia, PA: Taylor & Francis1999).
24.
BertC. W.JangS. K.StrizA. G., ‘Two new approximate methods for analyzing free vibration of structural components’ AIAA J.26 (1988): 612–618.
25.
BellmanR. E.CastiJ., ‘Differential quadrature and long-term integration’ J. Math. Anal. Appl.34 (1971): 235–238.
26.
BertC. W.StrizA. G.JangS. K., ‘Nonlinear deflection of rectangular plates by differential quadrature’ Computational engineering science (1988) ch. 23.
27.
StrizA. G.JangS. K.BertC. W., ‘Nonlinear bending analysis of thin circular plates by differential quadrature’ Thin-Walled Struct.6 (1988): 51–62.
28.
ShuC.RichardsB. E., ‘Application of generalized differential quadrature to solve two-dimensional Navier Stokes equations’ Int. J. Numer. Methods Fluids15 (1992): 791–798.
29.
BertC. W.MalikM., ‘Differential quadrature method in computational mechanics: A review’ Appl. Mech. Rev.49 (1996): 1–27.
30.
YokoyamaT., ‘Vibrations of Timoshenko beam-columns on two-parameter elastic foundations’ Earthq. Eng. Struct. Dyn.20 (1991): 355–370.
