The basis of this work is a novel symbiosis of mechanics of solids and spherical geometry to quantify and illustrate the variation of the -non-membrane- percentage of the strain energy in the prebuckling region of elastic structures. The zenith angle of an arbitrary point of a specific curve on an octant of the unit sphere, called buckling sphere, is related to this energy percentage. For the limiting case of buckling from a membrane stress state this curve degenerates to a point, characterized by zero values of both spherical coordinates. For all other stress states the azimuth angle increases with the proportionally increasing load. Its magnitude at the stability limit correlates with a quantity that depends on both the -non-membrane- deformations and the stiffness of the structure at incipient buckling. The azimuth angle is computed with the help of the so-called Consistently Linearized Eigenproblem (CLE), which is solved by means of the Finite Element Method (FEM). This eigenvalue problem is the basis for a geometrical hypothesis for the -non-membrane- percentage of the strain energy. Implementation of several routines into the finite element sofware MSC.MARC allows for its alternative computation in a conventional manner. In the theoretical part of the work, the concept of the buckling sphere is presented. Thereafter, its numerical realization is described. MSC.MARC permits taking advantage of the versatility of commercial FE software. Simultaneously it allows for modifying the iterative solution process within the framework of the FEM. With the help of the software MATLAB, the derivatives of the global tangent stiffness matrix with respect to a dimensionless load parameter are computed. An iterative procedure is used to increase the accuracy of the solution of the CLE. A further improvement of the numerical solution is achieved by means of examining the numerical precision of the output of MSC.MARC. Automatization of the computation is accomplished with the help of PYTHON. The subsequent numerical investigation consists of several examples, referring to buckling from a membrane stress state, a pure bending stress state, and a general stress state. The practical motive for this research is the intention to investigate the influence of -non-membrane- action just before buckling on the initial postbuckling behavior of elastic structures.