This article illustrates the numerical solutions of one-dimensional (1D) consolidation of saturated soils under both time-dependent aperiodic and periodic loadings considering single drainage boundary conditions. The numerical analysis involves employing the Galerkin Method of Weighted Residual (GMWR) for the development of Finite Element (FE) formulation of 1D consolidation problem to obtain the basic matrix equations with an aim to derive approximate solutions. Temporal discretization is achieved through a fully implicit method, while spatial variables are subjected to discretization employing identical shape functions. The uncertainty of the coefficient of consolidation (cv) is addressed by taking a set of values following a normal distribution in the range of 0.6–1.2 m2/yr. Realistic excess pore water pressure (EPWP) profiles of 1D consolidation under time-dependent aperiodic and periodic loading conditions are achieved by incorporating the randomness of cv. This contrasts with conventional assumptions and leads to more realistic EPWP. The proposed numerical solution demonstrates exceptional alignment with the analytical solution, and the accuracy of the proposed method is affirmed by a significantly reduced mean square error when compared to values of EPWP derived from the analytical formulation. Additionally, analytical solutions for Biot’s poro-elasticity equations under periodic loading are developed, considering single drainage boundary conditions. Furthermore, the FE formulation for the 1D representation of Biot’s quasi-static (QS) theory is outlined, employing the Principle of Virtual Work (PVW). Subsequently, profiles depicting displacement and pore pressure against normalized depth are graphically illustrated. The FE solutions are then compared with the analytically derived solutions, revealing a remarkable congruence in the profiles of displacement and pore pressure, affirming the accuracy of the presented results. The study also explores the impact and uncertainty of different soil and loading parameters, including permeability, Young’s modulus, and applied loading frequency on the displacement and pore pressure profiles. Examining random variations in soil and loading parameters, as opposed to traditional analyses produces practical solutions for displacement and pore pressure. This not only enhances the realism of outcomes but also provides valuable insights into the soil’s response under the specified periodic loading conditions.