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IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 1
D. M. OLIVEIRA | N. A. SILVA | C. F. BREMER | H. INOUE
According to Silva [9], if the
B
2
coefficient does not exceed the value
of 1.1 on all storeys, the structure can be considered almost insen-
sitive to horizontal movement and, in this case, global second order
effects can be ignored. When the greater
B
2
is situated between
1.1 and 1.4, the approximate
B
1
-B
2
method can be utilized for the
bending moment, with the other efforts (axial and shearing forces)
being directly obtained from the first order analysis. Lastly, when
B
2
> 1.40, the recommendation is that a rigorous second order
elastoplastic analysis be performed. Silva [9] also adds that, in the
event 1.1 <
B
2
≤
1.2, the bending moments can alternatively be
based on a first order analysis performed with the magnified hori-
zontal efforts by the greater
B
2
.
So it can be seen that, like the
g
z
coefficient, the
B
2
coefficient is
an “indicator” of the importance of global second order effects on
a structure. In this way, in the next item, an expression capable of
relating these parameters will be obtained.
4. Relation between coefficients
g
z
and
B
2
Figure [1] shows a structure consisting of three storeys of equal
length (
L
). In this figure the vertical (
P
id
) and horizontal (
F
hid
) design
forces working on each storey
i
, along with their respective hori-
zontal displacement (
u
i
) are also shown.
To calculate
g
z
, equation (1), the values of
M
1,tot,d
and
∆M
tot,d
need to
be determined. Through equations (2) and (3), we get, respectively:
(6)
uating second order effects by multiplying first order moments by
g
z
is
based on the assumption that the successive elastic lines produced
by vertical force action on the structure with displaced joints follow in
geometric progression. Indeed, it was seen in countless cases that up
to the value
g
z
= 1.3 this assumption is valid with less than 5% error.
However, there are some particular situations where the assumption
formulated in developing the method does not apply or applies with
greater errors. As examples of these exceptional cases, Vasconcelos
[6] quotes: when there is a sudden change in inertia between stories
(in particular between the ground and first floor), where ceiling heights
from one floor to the next are very different, cases of column transition
in beams, when there is torsion in the spatial frame or uneven settling
in the foundations, and others.
Oliveira [7] did an evaluation of the
g
z
coefficient’s efficiency as a first
order efforts magnifier (for bending moments, axial and shearing
forces) and as a horizontal loads magnifier, to obtain final, including
second order, efforts. The study was carried out for structures with
maximum
g
z
values in the region of 1.3, that is, for which, according
to NBR 6118:2007 [2], the simplified final efforts evaluation process
employing the
g
z
coefficient is still valid. It was found that the
g
z
coef-
ficient must be utilized as magnifier of first order moments (and not
for horizontal loads) to obtain final moments. In the case of axial
force on columns and shearing force on beams, magnification by
the
g
z
coefficient was not necessary, since the first and second order
efforts values obtained in these cases were practically the same.
3. Coefficient
B
2
To evaluate second order effects on steel structures, AISC/LRFD
[8] adopts the approximate method of amplifying the first order
moments by magnification factors
B
1
and
B
2.
So the second order
bending moment,
M
Sd
, must be determined by means of the follow-
ing expression:
(4)
M
Sd
= B
1
M
nt
+ B
2
M
lt
M
nt
being the design bending moment, assuming there is no side
sway in the structure,
M
lt
being the design bending moment due to
the frame’s side sway; both
M
nt
and
M
lt
are obtained by first order
analyses. The
B
1
amplification coefficient
depicts the
P-δ
effect,
relating to the instability of the bar, or to local second order effects;
B
2
considers the
P-Δ
effect, relating to the instability of the frame,
or to global second order effects.
The
B
2
coefficient can be calculated for each storey of the structure, as:
(5)
Sd
Sd
h0
2
H
N
L
1
1
B
with
ΣN
Sd
as the summation of the design axial compression forces
on all the columns and other elements resistant to the storey’s ver-
tical forces;
∆
0h
as the relative horizontal displacement;
L
as the
storey’s length and
ΣH
Sd
as the summation of all the design hori-
zontal forces on the storey producing
∆
0h
.
Figure 1 – Three-storey structure subjected
to vertical and horizontal forces
L
L
L
3
u
2
u
1
u
P
P
P
F
h3d
h2d
F
F
h1d
3d
2d
1d