A Quadrilateral Membrane Hybrid Stress Element with Drilling Degrees of Freedom

A kind of quadrilateral membrane assumed stress hybrid finite element with drilling degrees of freedom — which contains a traction-free inclined side — has been developed based on a modified Hellinger-Reissner principle. The stress equilibrium conditions are introduced through the use of additional displacements as Lagrange multipliers. The assumed stresses are expressed in complete polynomials in natural coordinates and a rational procedure is to choose the displacement terms such that the resulting strains are also of complete polynomials of the same order. The combination of the special elements with the ordinary assumed displacements elements can be efficiently used for analyzing the stress concentrations around some cutouts. The stress distribution of solid with symmetric V-shaped notches in different angles is analyzed to evaluate the following three cases: 1) Combining the special assumed stress hybrid elements with the ordinary isoparametric elements which are derived by the conventional assumed displacement approach; 2) Using the conventional assumed displacement elements with drilling degrees of freedom everywhere; 3) Using the conventional assumed displacement isoparametric elements everywhere. Numerical results have demonstrated that the special element can provide much more accurate stress concentration factors and the distributions of circumferential stress along the rim of the notches than those obtained by using other methods.