Basic HTML Version
76
IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 1
Considerations about the determination of
g
z
coefficient
1. Introduction
Of late, erecting more economical, slender structures and taller,
bolder buildings has become increasingly common.
The taller and more slender the building, the greater the strains
present, particularly those resulting from lateral actions. In these
cases the stability analysis and evaluation of second order effects
start taking on fundamental importance in the structural project.
Second order effects arise when the structure equilibrium consid-
ering the deformed configuration study is done. In this way, exist-
ing forces interact with displacements, thereby producing addition-
al efforts. Second order efforts introduced by the structure joints
moving horizontally, when subject to vertical and horizontal loads,
are referred to as global second order effects.
It is well known that all structures are displaceable. However, hori-
zontal joint displacements are small in some more stiff structures
and, as a result, second order global effects have little influence on
total efforts, and so can be ignored. These structures are referred
to as nonsway structures. In these cases, bars can be sized sepa-
rately, with their extremities tied, where efforts obtained by the first
order analysis are applied.
On the other hand, some more flexible structures have significant
horizontal displacements and therefore global second order effects
depict an important part of final efforts and cannot be ignored. This
is the case of sway structures for which a second order analysis
must be done.
According to NBR 6118:2007 [2], if global second order effects are
less than 10% of the respective first order efforts, the structure can
be classified as being nonsway structure. Otherwise (that is, when
global second order effects are over 10% higher than first order ef-
fects), the structure is classified as being sway structure.
NBR 6118:2007 [2] also establishes that structures can be classi-
fied using two approximate processes, the
α
instability parameter
and the
g
z
coefficient. However, the
g
z
coefficient goes beyond the
α
parameter, since it can also be utilized to evaluate final efforts,
which include second order efforts, as long as their value does
not exceed 1.3. However, it is obvious that, for second effects to
be evaluated satisfactorily, the
g
z
coefficient needs to be calcu-
lated accurately.
It is worth noting that the
g
z
coefficient must be employed in
reinforced concrete structures. To assess second order effects
on steel structures, the
B
2
coefficient must be utilized. As with
g
z
, this coefficient is also able to provide an estimate of a struc-
ture’s final efforts, as long as their value does not go beyond a
certain threshold.
Within this context, this paper’s primary intention is to ascertain
the adopted structural model’s influence in calculating the
g
z
co-
efficient. Thus, the
g
z
values for two medium height reinforced
concrete buildings are determined, considering five distinct three-
dimensional models developed utilizing ANSYS-9.0 [1] software.
The results obtained make it possible to identify the more ade-
quate models for putting the project into practice, as well as those
whose utilization could prove disadvantageous and uneconomical.
Moreover, the attempt has been made to carry out a comparative
study of coefficients
g
z
and
B
2
. To conduct the study, first of all an
expression associating these parameters is developed. Next, the
g
z
and
B
2
values for several medium height reinforced concrete build-
ings are calculated, utilizing ANSYS-9.0 [1] software.
2. Coefficient
g
z
NBR 6118:2007 [2] ordains that the
g
z
coefficient, valid for reticu-
lated structures at least four stories high, can be determined from a
first order linear analysis, by reducing the structural elements’ stiff-
ness, in order to consider the physical non-linearity approximately.
For each load combination, the
g
z
value is calculated using the fol-
lowing expression:
(1)
d tot
d tot
z
M
M
, ,1
,
1
1
- M
1,tot,d
(first order moment) being: a sum of the all the horizontal
force moments (with their design values) of the considered combi-
nation relative to the structure base, which can be written as:
(2)
M
1,tot,d
=
(F
hid
h
i
)
F
hid
being the horizontal force applied to storey
i
(with its design
value), and
h
i
being the height of storey
i.
-
ΔM
tot,d
(increase in moments after the first order analysis) being: a
sum of the products of all the vertical forces working on the struc-
ture (with their design values), in the considered combination, by
the horizontal displacements of their respective application points:
(3)
ΔM
tot,d
=
(P
id
u
i
)
P
id
being the vertical force working on storey
i
(with its design val-
ue), and
u
i
being the horizontal displacement of storey
i.
Bearing in mind that second order effects can be ignored as long
as they do not show a greater than 10% increase in the respective
first order efforts, a structure may be classified as being nonsway
structure if its
g
z
≤
1.1.
NBR 6118:2007 [2] establishes that final efforts (first order + sec-
ond order) can be evaluated from the additional 0.95
g
z
horizontal
efforts magnification of the considered loading combination, as
long as
g
z
does not exceed 1.3. However, according to the NBR
6118:2000 [3] Revision Project, final efforts values could be ob-
tained by multiplying the first order moments by 0.95
g
z
, also on
the condition that
g
z
≤
1.3. It is therefore understood that
g
z
ceased
to be the first order moment magnifier coefficient and became the
horizontal loads magnifier coefficient.
According to Franco &Vasconcelos [4], utilizing
g
z
as a first order
moments magnifier provides a good estimate for the second order
analysis results; the method was applied successfully on tall build-
ings with
g
z
in the region of 1.2 or more. Vasconcelos [5] adds that
this process is valid even for
g
z
values lower than 1.10, in which
cases technical norms allow second order effects to be disregarded.
It is also noted that, according to Vasconcelos [6], the process of eval-