A closed form of moment-curvature relation is derived for fibers with nonlinear material characteristics in the power law form. Closed form of bending rigidities and shape factors are obtained for rectangular, elliptical, and hollow elliptical cross sectional fibers. Bending stress-strain relation is also derived for elliptical cross sectional fibers made of nonlinear material characteristics. It is possible to obtain both a closed form solution for small deflection and a numerical solution for large deflection of beams subjected to various load conditions. To illustrate the use of derived relation, large deflection of cantilever beam defined by material characteristics in the power-law form under combined load obtained by using a numerical method.