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1. Introduction
1.1 Second order effects in building structures
Depending on its flexibleness, a building bracing structure, when
simultaneously subject to gravity and wind loads, may develop addi-
tional effects to those usually obtained in a first order linear analysis
(in which the equilibrium is verified in the non deformed structure).
They are the second order effects, in whose computation the mate-
rial nonlinear behavior (physical nonlinearity) and the structure de-
flected shape (geometric nonlinearity) must be considered.
The work of Beck and König [1], brought in 1967, represented an im-
portant advance in the development of tall buildings global stability
analysis. A very easy criterion to apply was established, determining
that the second order effects may be neglected, provided that they
don’t represent an increase more than 10% on the first order effects.
Figure 1 shows the simplified model for the bracing system. At first,
all bracing substructures are grouped in a single column, while all
braced elements (bearing elements that don’t belong to the bracing
system) are replaced by an assemblage of hinged bars, as shown
in figure 1-a. The wind is considered by means of a
w
rate uniform
load.
P
and
V
are the floor vertical loads, applied on the bracing sub-
structures and braced elements, respectively. The loads
w
,
P
and
V
are considered with their characteristic values. Thereafter, in order
to make possible to determine the second order effects by means of
a continuum analysis, an equivalent approximate model, shown in
figure 1-b, is adopted, with a continuous and uniform distribution of
floors and vertical loads (
p
=
P/h
e
v
=
V/h
).
Concerning to the influence of the loads
V
, acting on the braced
elements, Beck and König [1] proved that, when the system dis-
torts laterally, horizontal forces are transmitted through the floor
members to the bracing system, increasing the bending moment
on its support. It can be proved that this increase is given by the
sum of the forces
V
multiplied by the horizontal displacements of
the respective floors. Therefore, in order to compute this bending
moment including second order effects, the vertical loads acting
on the bracing system would be given by the sum of its proper
P
loads and
V
loads.
In 1978, the criterion proposed by Beck and König [1] was included
in the
Comité Euro-International du Béton
recommendations (CEB
[2]). Its application consists in comparing the global bending mo-
ments at the bracing system support
M
I
(considering only first or-
der effects) and
M
II
(including second order effects):
(1)
I
II
M M
1,1
£
or
(2)
2
75111
8
7511
1
2
751
2
3
2
tot
tot
tot
wH ,
,
EJ
v)H (p ,
wH ,
´ £
+
-
×
It can be noted that
M
I
and
M
II
are due to factored loads, since
the rates
w
,
p
and
v
are multiplied by 1,75. On the other hand,
the physical nonlinearity is regarded taking
EJ
= 0,7
E
cm
J
for the
structural members, where
E
cm
J
represents the sum of the bracing
substructures stiffness coefficients at the non cracked stage. Thus,
performing this substitution leads to the condition:
105
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
R. J. ELLWANGER
Figure 1 – Simplified model for the bracing system
A
B