doi: 10.18720/MCE.77.3
Finite element models in stresses for plane elasticity problems
Yu.Ya. Tyukalov,
Vyatka State University, Kirov, Russia
The solution of
the plane problems of elasticity theory on the basis of stress
approximation is considered. To construct the solution, the additional
energy functional is used. With the help of the principle of possible
displacements, algebraic equations of equilibrium of the nodes of the
grid of finite elements are constructed. Equilibrium equations are
included in the functional of additional energy by means of Lagrange
multipliers. The necessary relations for rectangular and triangular
finite elements are obtained. Variants with constant and
piecewiseconstant approximations of stresses in the region of the
finite element are considered. The ribbon width of system of the
solving linear equations is estimated. Calculations have been made for
the bended beam and for stretched plate with the hole, for the
different grids of finite element. It is made comparison of the
solutions obtained in stresses with the solutions obtained by finite
element method in displacements and with exact solutions. It is shown,
that for plane problems in the theory of elasticity, solutions based on
stress approximations make it possible to obtain convergence of
displacements to exact values from above. For coarse grids, solutions
based on piecewise constant stresses much more accurate results, but
require large computational costs, since the width of the ribbon of
nonzero elements of the resolving system of linear algebraic equations
is approximately twice as large as in the other considered variants.
Finite elements models in stresses allow constructing solutions, which
are alternative to solutions obtained by finite element method in
displacements.
Keywords:
finite elements; stress approximation; functional of additional energy; Lagrange multipliers
Read the whole article in pdf
(Tyukalov Yu.Ya. Finite element models in stresses for plane elasticity problems. Magazine of Civil Engineering. 2018. No. 1. Pp. 23–37. doi: 10.18720/MCE.77.3.).
