Column and Beam Frame Element Formulation

A frame element is a typical 1D finite element with six degrees of freedom (DOF) at each end: three translational and three rotational degrees of freedom. As in RAM Frame, shear deformations are incorporated into element stiffness matrix. The element formulation is based on linear theory where only axial forces, moments and torsions are considered and second order effects and coupling between these element forces are not accounted for. Refer to the RAM Frame manual for specifics on the matrix formulation that is implemented.

For unsymmetrical cross sections, it is assumed that loads are applied through shear center of the cross-section. For unsymmetrical T -sections and L sections, it is further assumed that the members are continuously supported along their lengths so as to prevent twist. For beams cast monolithically with the slab this is a justified assumption.

Wall Element Formulation

A four-node finite element for modeling walls is used in RAM Concrete. The element consists of six degrees of freedom at each of the four nodes: three translational and three rotational.

The wall stiffness matrix includes out-of-plane bending stiffness, which is not considered in RAM Frame. Previously, a finite number (in the order of 10^-6) is introduced into the formulation to prevent any numerical singularity for out-of-plane bending. However, in this current formulation, a robust calculation for out-of-plane bending is considered.

The membrane stiffness for in-plane bending is calculated in a similar way to RAM Frame. However, it is augmented with drilling degrees of freedom. (The current formulation is similar to the developments given in References 1-3 in Section 3.7). This new approach differs from previous wall formulation used in RAM Frame in which internally a rigid beam is introduced into the formulation to provide moment continuity in the major direction (see the RAM Frame manual).

Loads are applied along the edges of walls, which are handled in a similar way in frame elements: they are resolved into their perpendicular and axial components.

Slab Deck Element Formulation

Two-way slab decks are meshed and represented using shell finite element for analysis. The implemented element is a quadrilateral shell element and it has a total of six degrees of freedom at its each node: three translational, two bending and one drilling. The formulation for element stiffness matrix includes membrane stiffness and plate bending stiffness. These stiffness matrices are calculated separately and then combined to form element stiffness matrix for the shell element (see the RAM Frame manual for further details).

The membrane stiffness accommodates three degrees of freedom at each node, namely two in-plane translational and one drilling degrees of freedom. The formulation utilizes Allman type shape functions within Hughes-Brezzi variational formulation framework. It includes correction matrix to remove any existing membrane locking from element behavior. Also, another correction matrix is applied to calculated stiffness matrix in case of warped planes of shell (Ibrahimbegovic, at. al., 1990; Taylor, 1987; and Long at. al., 2004).

The plate bending stiffness is derived based on thin plate assumption and it is a typical Discrete Kirchoff Element (shear deformations over the thickness of the element ignored). The formulation includes three degrees of freedom at each node: remaining two rotational degrees of freedom and one translational degrees of freedom, perpendicular to the plane of the shell.

Unlike beam, brace and column elements whose stiffness coefficients are integrated exactly, the shell finite element gives results that are "exact" or close to the theoretical solution only if a finer mesh is used and convergence achieved. For most practical purposes, the use of only a few slab deck element in a floor may significantly underestimate deflections. In some cases, it could also give wrong results. RAM Concrete automatically meshes two-way slab deck and one-way slab decks if included in a slab edge loop with two-way slab deck.