Civil Engineering Reference
In-Depth Information
A
S
= 100
mm
2
Concrete (f
c
= 26.7
MPa)
15
mm
1000
mm
A
S
= 50
mm
2
Steel (f
c
= 474
MPa)
Figure 4.20
Modelling of rigid diaphragms in framed structures: 3D fi nite element model of SPEAR frame without
(
top
-
left
) and with rigid (
top
-
right
) diaphragms and horizontally rigid member used to simulate the diaphragmatic
action (
bottom
)
The assumption of rigid fl oors is economical in the analysis since several degrees of freedom can be
condensed and the order of the stiffness matrices reduced. For example, the three-storey and two- bay
irregular frame shown in Figure 4.20 possesses 162 DOFs: at each storey, the slab connects 9 nodes
with 6 DOFs (three translations and three rotations).
If the three diaphragms of the SPEAR frame in Figure 4.20 are rigid, in-plane displacements of all
nodes, defi ned as 'slave nodes', can be expressed, through principles of basic mechanics, as a function
of the corresponding DOFs of a 'master node', at each storey level. Consequently, the total number of
DOFs is reduced from 162 to 90. These include two translations (in the plane of the diaphragm) and
one rotation (about the
z
-axis) for the master nodes and one translation (along
z
- axis) and two rotations
for each slave node. Master nodes are generally assumed to coincide with the mass centre of the
diaphragm.
In several commercial software packages, in-plane rigidity of horizontal diaphragms is modelled
automatically, e.g. by means of diaphragm constraints or constraint equations. In these cases, master
nodes possess 3 DOFs at each storey level: the translations along
x
- and
y
-direction and a rotation about
the
z
- axis. The three out -of-plane DOFs are often not kinematically constrained. Rotational DOFs about
x
- and
y
-axes can be neglected. Alternatively, diaphragm effects can be simulated by inserting horizon-
tally rigid members in the slab (e.g. Jeong and Elnashai, 2005). In the sample frame in Figure 4.2, in
order to model rigid diaphragms, each corner of the slabs is diagonally connected to the opposite corner.
The dimensions and reinforcement of connecting RC members shown in Figure 4.20 are such that the
additional members do not provide duplicate stiffness to the fl exural resistance of beams. The contribu-
tion of slabs to fl exural stiffness of beams is already modelled by effective width of T-beam models.
The thickness of the connecting elements is determined by iteration such that the contribution of con-
necting elements to the vertical stiffness is negligible, while the contribution to the horizontal stiffness
remains signifi cant. Figure 4.21 shows the angle of torsion at all corners of slabs that are located at
column points for the fi rst fl oor of the SPEAR frame. The analytical model with additional connecting
