Analytical Solutions for Static Shear Correction Factor of Functionally Graded Rectangular Beams

Most practical analyses of functionally graded beams, particularly in aerospace, aircraft, automobile, and civil structures, are based on the first-order shear deformation theory. However, a key factor in practical application of the theory is determination of the transverse shear correction factor, which appears as a coefficient in the expression for the transverse shear stress resultant. The physical basis for this factor is that it is supposed to compensate for the assumption that the shear strain is uniform through the depth of the cross section. Using the energy equivalence principle, a general expression is derived for the static shear correction factors in functionally graded beams. The resulting expression is consistent with the variationally derived results of Reissner's analysis when the latter are reduced from the two-dimensional (plate) case to the one-dimensional (beam) one. The beams are assumed to have an isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law, or sigmoid distribution in terms of the volume fractions of the constituents. A convenient algebraic form of the solution is presented and an example is given to illustrate the use of the present formulation. Numerical results are presented to show the effect of the material distribution on the shear correction factor for various functionally graded beams. Further, a comparison of the results of power-law, or sigmoid functionally graded materials is investigated.

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