Civil Engineering Reference
In-Depth Information
Fibre
Layer
B
B
Figure 4.14
Discretization of cross sections: fi bre (
left
) and layer (
right
) model
4.5.3 Components and Systems for Structural Modelling
The evaluation of seismic response of structures by FE analysis requires that system components, such
as beam-columns, braces, slabs and walls are modelled using discrete elements spanning between nodes.
Below is a summary of commonly used elements for earthquake response analysis:
i .
Beams
: Beam elements are used for both planar and spatial models of structures to represent
beams and columns. In 2D analysis, they possess 3 DOFs per node: 2 displacements and a
rotation about the axis perpendicular to the plane of the structure. Axial loads, bending moments
and shears are the internal actions. Three-dimensional beams have 6 DOFs per node, these are
3 displacements and 3 rotations. Internal actions include axial load, 2 shear forces, 2 bending
moments and torsion. Beam elements are commonly implemented in FE programs as two- noded
elements. Higher-order formulations are, however, also available (e.g. Cook
et al
., 2002 ). The
higher-order beams include one or two additional nodes at intermediate locations along the
element length. The number of nodes of beam elements depends on the types of polynomial
used as shape functions. Euler or Timoshenko formulations are used depending on the geometric
characteristics of the beam element. For deep and stocky members (aspect ratios less than ∼ 3 - 4),
shear deformations are signifi cant and it is more accurate to employ Timoshenko's beam theory.
Similarly, torsional deformations should be accounted for when employing elements with open
sections. Cubic elastic-plastic 3D beam-column elements in Zeus-NL (Elnashai
et al
., 2003 )
may be used to model RC frame members of the SPEAR building in Figure 4.2. This cubic
element was formulated by Izzuddin (1991) and is used for detailed inelastic modelling, making
use of the uniaxial inelastic material models for steel and concrete described in Section 4.5.1 .
It accounts for the spread of inelasticity along the member length and across the section depth.
Geometrical non-linearities are accommodated using a Eulerian formulation. Beam elements
can also be used to model structural components subjected only to axial forces, such as braces
and cables. These elements are known as ' rods ' or ' bars ' . In some FE computer programs, rods
are derived from beam elements by releasing all but the axial DOFs;
ii.
Plates and shells:
These include triangular and rectangular elements. Isoparametric formulations
permit quadrilateral elements to have non-rectangular shapes and mild curvature or irregular
geometry. The number of nodes of plate and shell elements depends on the types of polynomial
used as shape functions, as in the case of beams. Lower- and higher- order formulations have
been implemented for these FEs (e.g. Cook
et al
., 2002). Following the general plate theory,
plate elements are assumed to have 3 DOFs per node: translation perpendicular to the plate and
rotations about two perpendicular axes in the plane of the plate. The DOFs per node for shells
