The solution procedure is split into several steps including localization of the global stiffness matrix while taking into account the support conditions (fixed or spring supports at joint or along lines, elastic subsoil), setting up the load vector and analysis of the system of equations using the Gaussian method with the Cholesky decompositions of the global stiffness matrix, which in this case is symmetric and band. The values of primary variables wz, φx a φy calculated at mesh nodes are then used to determine the internal forces mx, my, mxy, vx and vy together with the derived quantities m1, m2 and the values of reactions developed in supports.
The quality of results of the slab problem derived using the finite element method is strongly influenced by the type of slab element. The present formulation exploits a deformation variant of the finite element method to derive triangular and quadrilateral elements denoted as DKMT and DKMQ (Discrete Kirchhoff-Mindlin Triangle a Quadrilateral).
Formulation of the slab element implemented in the program is based on the discrete Kirchhoff theory of bending of thin slabs, which can be considered as a special case of the Mindlin plate theory developed upon the following assumptions:
- compression of slab in the z-direction is negligible compare to the vertical displacement Wz
- normals to the mid-plane of the slab remain straight after deformation but not necessarily normal to the deformed mid-plane of the slab
- normal stress σz is negligible compare to stresses σx, σy
DKMT and DKMQ elements have 9 and 12 degrees of freedom, respectively - three independent displacements at each node:
deflection in the direction of z-axis
rotation about x-axis
rotation about y-axis
The elements satisfy the following criteria:
- the stiffness matrix has correct rank (no zero energy states are generated)
- fulfill the patch test
- are suitable for the analysis of both this and thick slabs
- they show good convergence properties
- not computationally expensive
In case of well generated mesh the quadrilateral elements are preferable as show better behavior compare to triangular elements.
The slab can be reinforced by beams formulated on the basis of one dimensional beam element with embedded torsion and is compatible with slab elements (details can be found in literature). The primary variables are Wz, φx and φy and corresponding internal forces are M1, M2 and V3 (twisting and bending moments and shear force). The beam is characterized by the moment of inertia It a I2 (torsion, bending), area A and shear area As. These parameters can be calculated by the program based on the type of cross section. The analysis constructs 6x6 local stiffness matrices subsequently localized in to the global stiffness matrix of the structure.
I. Katili, A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields - part I: An extended DKT element for thick-plate bending analysis, Int. J. Numer. Meth. Engng., Vol. 36, 1859-1883 (1993).
I. Katili, A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields - part II: An extended DKQ element for thick-plate bending analysis, Int. J. Numer. Meth. Engng., Vol. 36, 1885-1908 (1993).
Z. Bittnar, J. Sejnoha, Numericke metody mechaniky, CVUT, Praha, 1992.