Civil Engineering Reference
In-Depth Information
These simple analyses are
first order
. That is, they neglect any increase
in the assumed sway
caused by the deformations of the structure under
load. Analyses that take account of this effect are referred to as
second
order
. A simple example is the elastic theory for the lateral deformation
of an initially crooked pin-ended strut.
φ
5.4.2
Elastic stiffnesses of members
The determination of these properties requires consideration of the
behaviour of joints and of creep and cracking of concrete. Creep in beams
will be allowed for by using a modular ratio
n
20.2, as before. For
columns, EN 1994-1-1 gives the effective modulus for concrete as
=
2
n
0
=
E
c
=
E
cm
/[1
+
(
N
G,Ed
/
N
Ed
)
ϕ
t
]
(5.7)
where
N
Ed
is the design axial force,
N
G,Ed
is the part of
N
Ed
that is per-
manent, and
ϕ
t
is the creep coefficient. Expressions for the stiffnesses of
composite columns are given in Section 5.6.3 in terms of
E
c
. Typical
values for the short-term elastic modulus
E
cm
are given in Section 3.2.
For cracking it will be assumed, from Section 4.3.2, that the 'cracked
reinforced' section of each beam is used for a length of 15% of the span
on each side of the central column. The joints are assumed to be rigid
at the central column and nominally-pinned at the external columns, as
explained earlier.
5.4.3
Method of global analysis
The condition of EN 1993-1-1 for the use of first-order analysis is
α
cr
α
cr
is the factor by which the design loading would have
to be increased to cause elastic instability in a sway mode.
There is a well-known hand method of calculation of
≥
10, where
α
cr
for simple
frames involving
s
and
c
functions, which have been tabulated [50]. Com-
puter programs are available for more complex frames. For beam-and-
column plane frames in buildings, EN 1993-1-1 gives the approximation
α
cr
=
(
H
Ed
/
V
Ed
) (
h
/
δ
H,Ed
)
(5.8)
to be satisfied separately for each storey of height
h
. In this expression,
V
Ed
and
H
Ed
are the total vertical and horizontal reactions at the bottom
of the storey, with forces
N
φ
included in
H
Ed
, and
δ
H,Ed
is the change in
lateral deflection over the height
h
.
It will be shown later that, for the frame in Fig. 5.1, the lateral stiffness
of the floor slabs is so much greater than that of the columns that almost
