Postbuckling analysis of edge cracked functionally graded Timoshenko beams under end shortening

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Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler-Bernoulli beams (2013, background): “[8,9] as well as Yang and Chen [10] studied free vibrations, buckling and post-buckling of FGM TBs containing open cracks by assuming an exponential variation of material properties in the thickness direction.”
A higher-order theory for static and dynamic analyses of functionally graded beams (2010, background): “Recently, buckling, flexural vibration and postbuckling of cracked functionally graded Timoshenko beams have been studied in [ 16 ,17].”
An analytical study on the nonlinear vibration of functionally graded beams (2010, unknown): “Introducing the following dimensionless quantities [28, 29]”
Dynamic Stability of Functionally Graded Carbon Nanotube-Reinforced Composite Beams (2013, background): “investigations on the static bending [21, 22], elastic buckling [23, 24], dynamic stability [24–29], postbuckling [30, 31], linear and nonlinear free vibration [32–37] of FGM beams, plates, and shells have been conducted for the past 20 years.”
Nonlinear forced vibration analysis of clamped functionally graded beams (2011, methodology): “We introduce the following dimensionless quantities [26, 27 ]:”