Paraboloid shells are translational shells formed from two parabolas placed at the rightrnangle to each other and by the translation of one parabola over the other. Ellipticalrnparaboloid is formed by two unequal parabolas with the same curvature sign. rnThin concrete elliptic paraboloid shell with large spans are susceptible to buckling.rnDetermining the critical buckling load, internal forces and free vibration is difficultrnbecause of the lack of experimental study on such shell type. rnThe research is focused on the analysis of an elliptic paraboloid shell structure under itsrnown weight. Geometrical and material linearity are considered in modeling and analysisrnof the structure in this study. The objective of the research is the parametric study onrnstability analysis of a thin shell structure under its own weight. rnIn this study, the internal forces, moments, deformation, stability, and free vibration ofrnelliptic parabolic shell analysis are performed by theoretical and finite element methods.rnTheoretical analysis is done by shallow shell theory and the finite element analysis isrncarried out by using DLUBAL RFEM 5.3.1. rnTheoretical formulas in this study are coded by the MATLAB program. The codes arernused to determine internal force, moments, in-plane shear, displacement, critical load andrntime, angular frequency, natural frequency in free vibration are written in the appendix. rnThe influences of the different parameters on the critical loads of the shells are therebyrnstudied. The parameter considered is thickness, the radius of curvature, modulus ofrnelasticity and side rise of the shell to determine the critical buckling load. Free vibrationrnanalysis of rectangular and square planned elliptic paraboloid shell structures was done. rnFinally, the internal forces, displacement, critical buckling load, and free vibration analysisrnresult in the study presented graphically and discussed in detail. The result of the finiternelement analysis is then compared with the theoretical analysis results.