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| Analytical relationships between the solutions of buckling of functionally graded material (FGM) Timoshenko beams and those of the homogenous Euler-Bernoulli beams subjected to axial compressive load were investigated for different boundary conditions. Based on the first order shear deformation theory, governing equation of buckling of FGM beams with the material properties changing continuously arbitrarily in the thickness direction were derived only in terms of the deflection, which has the same form as that of the homogenous Euler-Bernoulli beams. By solving the governing equation with specific boundary conditions, an analytical relationship between the critical buckling loads of the FGM Timoshenko beams and those of the corresponding Euler-Bernoulli beams were obtained which is valid for the clamped-clamped(C-C), simply supported-simply supported (S-S) and clamped-free(C-F) end constraints. Unfortunately, the above mentioned relationship is not valid for the beams with clamped-simply supported (C-S) ends. However, a simple eigenvalue equation is presented to find the critical buckling load for a C-S FGM Timoshenko beam. Consequently, buckling loads of FGM Timoshenko beams with C-C, S-S and C-F ends can be simplified as calculations of two coefficients determined by the material gradients and the geometry of the FGM beams when the critical buckling load parameters are known.) |
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| Keywords:functionally graded material; Timoshenko beam; buckling; critical loads; analytical solution |
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