An analytical study is presented in this paper on the effect of material nonlinearity on buckling and postbuckling responses of fiber composite laminate plates and shells subjected to general mechanical loading. The material nonlinearity of the composite is modeled by power-law type, shear nonlinear constitutive equations. The nonlinear effective composite constitutive equations in an incremental form are incorporated into a geometrically nonlinear analysis for studying the composite buckling and postbuckling deformations. A modified Riks solution scheme along with an updated Lagrangian formulation is used to construct the equilibrium path during the composite postbuckling. Several numerical examples are given to illustrate the effect of material nonlinearity on critical loads and postbuckling deformations of the composite plates and shells under axial and pressure loading. Influences of lamination parameters and geometric imperfection on the postbuckling equilibrium path and strength of the composite structures with nonlinear shear properties are also shown and discussed.