Improving the Bergan Wang Quadrilateral plate element by using the UI approach for analysis of isotropic and FGM sandwich plates

An efficient shear and bending‐locking‐free quadrilateral plate element using a modified Hellinger‐Reissner functional and the Bergan free formulation

“…The constant curvature patch test is a simple test used for several decades, mainly by the computational structural engineering community, to verify the absence of shear locking in plate elements based on the RM theory (References 5,8,23,24,36, among others). A constant curvature (pure bending) state is defined by: $$$$ {\displaystyle \begin{array}{l}w=-\frac{1}{2}\left({a}_1{x}^2+{b}_1{y}^2+{c}_1 xy\right)\\ {}{\upbeta}_x={a}_1x+\frac{1}{2}{c}_1y\\ {}{\upbeta}_y={b}_1y+\frac{1}{2}{c}_1x\end{array}} $$$$ with arbitrary constant values of a 1 , b 1 , c 1 .…”

“…• imposition of the energy-orthogonality condition between the linear and quadratic bending modes (like in the 'Bergan free formulation'). [22][23][24][34][35][36] • the use of static condensation at the element level to eliminate the increments of rotations { Δβ n } and the tangential shear strain…”

“…The main theoretical aspects of the formulation are: the use of a modified Hellinger‐Reissner functional where the transverse shear strains are the mixed independent variables. the enhancement of the rotation field using incomplete quadratic approximations of β x and β y (as for DKQ 18 and DKMQ 20 ) the representation of the mixed shear strains $$$ {\underset{\_}{\upgamma}}_x $$$ and $$$ {\underset{\_}{\upgamma}}_y $$$ as in MITC4 21 (linear in ξ and η) with constant tangential components on the four sides (also used in DKMQ 20 ). the kinematical shear strains (in terms of w , β x , β y ) are approximated consistently with the mixed shear strain components. imposition of the energy‐orthogonality condition between the linear and quadratic bending modes (like in the ‘Bergan free formulation’) 22–24,34–36 the use of static condensation at the element level to eliminate the increments of rotations $$$ \left\{\Delta {\upbeta}_n\right\} $$$ and the tangential shear strain $$$ \left\{{\underset{\_}{\upgamma}}_{s_n}\right\} $$$ components. all matrices are evaluated using a 2 × 2 Gauss integration scheme. …”

“…As a perspective of future contributions, we propose to evaluate the behaviour of BKWA for composite and FGM plates and compare the results with those obtained using other elements (see Reference 22,36,37). BKWA can also be extended to develop quadrilateral shell elements with a comparison with DKMQ 38 or MITC4+ 39,40 .…”