This paper presents a new nonlinear stiffness matrix of a finite eleme
nt without making any simplifications. This matrix inserts the quadrat
ic and cubic dependences of the unknown increments of generalized disp
lacements of nodes into the initially linearized system of equations.
For a bar element, the exact form of the matrix is defined and for ill
ustrative one-dimensional problems of pure tension the full system of
non-linear equations is given, which is solved by the method described
in [M. J. D. Powel, Hybrid Method for Non-linear Equations. McGraw-Hi
ll (1970)]. The calculation is evaluated as far as its convergence is
concerned, and the results are compared with those of a classical appr
oach to geometrically non-linear problems represented by the method of
finite elements. Thanks to the development of numerical methods of so
lving the systems of non-linear equations, the new non-linear stiffnes
s matrix can contribute significantly to a more efficient solution of
non-linear problems in the mechanics of elastic bodies.