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A micromechanics model is proposed for the elasticity of planar fiber networks (FNs). The FN is created by random deposition of linearly elastic straight rods within a region. The rods are bonded rigidly at contacts. Under external in-plane loading, the FN deformation consists of fiber bending, elongation, and contraction. An effective constitutive relation for fiber network is developed by averaging the strain energy dissipated by all possible fiber deformations in all directions. Numerical calculations are performed to analyze the effects of fiber aspect ratio and fiber concentration on the effective stiffness of the planar random FN. Finite element analysis (FEA) is performed and compared with the theoretical predictions of the effective FN moduli at several fiber concentrations. FEA results are in good agreement with theoretical predictions. The present model can be used for the prediction of mechanical properties, scaling analysis, and optimization of fiber assemblies.