Edge Nodes of Triangular Kirchoff Bending Thin Plate F.E.M.[Adini,Tocher​,BCIZ]

Bending triangular thin plate

The finite element analysis techniques are sometimes selected bending triangular finite element model. This models has main problem is continuity of element shape function for element geometric boundary conditions. Adini had selected first triangular bending element shape function ; w(x,y)=[1 x y x**2 y**2 x**3 x**2*y x*y**2 y**3] and he was neglect "x*y" polinomal parameter.
Nevertheless finite element has total nine degree of freedom and Pascal triangle description for shape function ten parameters. Smilarly Tocher was not neglect shape function "x*y" terms and he had collect üniform bending parameters "(x**2*y+x*y**2)".

This region second problem determine the analysis system pacth test. The Adini model has patch testing comforming for one dimensional bending analysis but two dimensional analysis non-conforming. Tocher model has non-conforming one and two dimensional bending analysis. Interesting
Some analysis is Sap2000(structral analysis program) made a patch testing error and this errors is very big.

Triangular plates analysis of this main problem had solution Bazely-Cheung-Irons-Zeinkeiewicz(BCIZ) Models. This solution is selected element shape function the homogen
element axis (e1,e2,e3) and conforming element per node patch testing for rigid body motion.
Per models are collate analysis for maximum displacements and rotations. Interesting , Adini and Tocher
Model , sap2000 and BCIZ models analysis results has smilarly values. Additional, Reissner-Mindline
Rectangular bending model is transformed triangular analysis. Because this model has isoparametric and if not need element node position than system is rigid. I think Sap2000 structural programs the smilarly technique.

This region finite element model & data

Triangular ares Homogen integration method (Gamma-Beta func.)
3Node-9Dof Bending Triangular Plate [Adini]
[1]Gauss-Quadratic integration method
[2]Mathlab 7.1 Symbolic tools

3Node-9Dof Bending Triangular Plate [Tocher]
[3]Gauss-Quadratic integration method
[4]Mathlab 7.1 Symbolic tools

3Node-9Dof Bending Triangular Plate [BCIZ]
[5]Gauss-Quadratic integration method
[6]Mathlab 7.1 Symbolic tools

Sap2000 (Structural analysis program) model BCIZ
Sap2000 (two dimensonal patch testing BIG ERROR)

4Node-12Dof Reissner Isoparametric plate Plate [BCIZ]
(Isoparametric plate convert triangular plate technique)