The definition of effective pressure with associated formula of the Bishop parameter for unsaturated porous medium proposed in the frame of the theory of macroscopic porous continuum has been controversial for a long time. This also affects the correct prediction of directly related generalized Biot effective stress. Based on the Voronoi cell model described with the discrete system composed of solid particles, binary bond liquid bridges and liquid films, the present paper presents the definitions of effective internal state variables at local material points in unsaturated porous continua with low saturation, i.e. effective pressure and generalized Biot effective stress. Using the proposed Voronoi cell model, their expressions are formulated with the information of hydro-mechanical meso-structure and moso-response evolved with incremental loading process exerted on the representative volume element (RVE) of unsaturated granular material. With the derived effective pressure formula, it is demonstrated that the effective pressure tensor of unsaturated porous continuum is anisotropic. It has not only an anisotropic effect on hydrostatic components, but also an effect on shear stress components, of generalized Biot effective stress tensor. It is demonstrated that the fundamental defect of both the generalized Biot theory and the so-called bivariate theory lies in that it is assumed that effective pore pressure tensor representing the hydro-mechanical effect of two immiscible pore fluids on the solid skeleton of unsaturated porous continua is isotropic. In addition, the Bishop parameter introduced as the weighted factor to define the isotropic effective pore pressure tensor is assumed not related to the matrix suction with very important effect on the hydro-mechanical response occurring at local material points over unsaturated porous continua. The derived formulae of both generalized Biot effective stress and effective pressure (including effective Bishop parameter reflecting the isotropic effect of effective pressure) can be upscaled to a local material point, where the RVE is assigned, in macroscopic unsaturated porous continua, for computational multi-scale methods represented by the concurrent computational homogenization method for unsaturated granular materials.