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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
A variable limit for the instability parameter of wall-frame or core-frame bracing structures
[2], on determining in its section 15 that the second order global
effects are negligible when lower than 10% of the respective first
order effects (fixed nodes structure). In order to “verify the possibil-
ity of dispensing the consideration of second order global efforts,
in other words, to define if the structure may be classified as a
fixed nodes one, without the need of a rigorous analysis”, ABNT [8]
presents two approximate procedures, based respectively on the
instability parameter and the
g
z
factor. The first one just consists of
the Beck and König [1] criterion application and determines that: “A
symmetrical framed structure may be considered as a fixed nodes
one, if its instability parameter a will be lesser than the
a
1
value,
according to the expressions:
(5)
)
/(
CSC k
tot
IE N H
=a
(6)
4
6,0
3
1,02,0
1
1
³ ' = Ù £ '
+ =
n
n n
a
a
“
n
is the number of horizontal bars levels (floors) above the founda-
tion or a slightly displaceable subsoil level.
H
tot
is the structure total
height, measured from the foundation top or a slightly displaceable
subsoil level.
N
k
is the summation of all vertical loads acting on
the structure (from the level considered for
H
tot
computation), with
their characteristic values.
E
CS
I
C
represents the summation of all
column stiffness values in the considered direction. In the case
of framed, trussed or mixed structures, or columns with variable
stiffness along the height, the
E
CS
I
C
value of an equivalent column
with constant section may be considered”. The determination of
this equivalence will be seen in section 2.1.
I
C
is the moment of
inertia considering columns gross sections.
E
CS
is the secant elas-
ticity modulus, expressed by:
(7)
2/1
5600
85,0
85,0
ck
Ci
CS
f
E
E
´ =
=
E
CS
, E
Ci
(tangent elasticity modulus) and
f
ck
(compressive charac-
teristic strength) are given in MPa. The NBR 6118 code also ad-
opted the Franco [5] propositions on determining different
a
1
val-
ues, depending on the bracing structure type: “The limit value
a
1
=
0,6, prescribed for
n
> 4, is generally applicable to the building
usual structures. It may be adopted for wall-columns assemblages
and rigid frames associated to wall-columns. It may be increased
until
a
1
= 0,7 in the case of bracing systems composed exclusively
by wall-columns and must be reduced to
a
1
= 0,5 if there are only
rigid frames.”
In a second order analysis, the effects of both physical and geomet-
ric nonlinearities must be considered. ABNT [8], in its item 15.7.3,
allows that the physical nonlinearity can be considered in an ap-
proximated manner, on calculating second order global efforts in
framed structures with four or more floors. This is done by means
of a reduction of the structural members (
EI
)
sec
stiffness factors as
(3)
0,54
)
/(
)
(
cm tot
tot
£
+
JE Hv p H
According to Vasconcelos [3], the results obtained by Beck and
König [1] could be applied only to building structures whose lat-
eral stiffness was concentrated in few columns rigidly connected
among themselves, in order to be considered as a single column.
The correspondence of this model with other types of bracing sys-
tems (variable section walls, rigid frames etc.) came to be done
through the equality of horizontal displacements due to horizontal
loads. The equivalent column would be that one with a stiffness
factor
EJ
such that the resulting horizontal displacements were
the same of the structure under consideration, for the same hori-
zontal loading. With the purpose of simplification, this equivalent
stiffness came to be determined based on the actuation of a unit
horizontal load at the building top. In Brazil, the procedure came
to be applied changing the load factor from 1,75 to 1,40 – see, for
example, Sussekind [4] – and came to be known as minimal stiff-
ness check. Consequently, inequality (3) changed to:
(4)
0,60
)
/(
)
(
I 28 S
tot
tot
£
+
-
J E Hv p H
where
E
S-28
is the concrete secant elasticity modulus at 28 days and
J
I
is the sum of bracing substructures inertias at non cracked stage.
In 1985, Franco [5] proposed that the equivalent column stiff-
ness have to be obtained based on the actuation of a uniformly dis-
tributed horizontal load, in place of the top unit load. Furthermore,
he preconized that the deflected shape of the bracing structure can
affect the Beck and König [1] criterion application. Thus, the coef-
ficient on the inequality (4) right hand would have its value defined
as a function of the bracing type:
- walls or cores: coefficient 0,7;
- wall-frame or core-frame structures: coefficient 0,6;
- only rigid frames: coefficient 0,5.
In 1995, Franco [6], dealing with the physical nonlinearity consid-
eration through structural members stiffness reduction, proposed
different reduction factors, specific for slabs, column members and
beam members with symmetrical and asymmetrical reinforcement.
Although not belonging to this work purpose, a mention deserves
to be made to the method based on the moment amplification fac-
tor
g
z
, presented in 1991 by Franco and Vasconcelos [7]. It also ap-
plies the criterion of 10% increase in relation to first order effects,
to define if a second order analysis is or not needed; however, in
this case it is done for each combination of horizontal and verti-
cal loads. Furthermore, under certain conditions, this method may
itself constitute a second order analysis. These features caused
this method to be rapidly disseminated and largely employed in
buildings structures design.
1.2 ABNT NBR 6118 prescriptions
The ABNT NBR 6118 [8], present Brazilian code for concrete struc-
tures design, adopted the fundamental idea presented in [1] and