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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
Thus, 88 different bracing systems were tested. Each test aimed
to determine the relation between vertical loads and horizon-
tal stiffness that would result in a 10% increase on the global
bending moment at building support, concerning to first order
analysis; in this way, the limit
a
1
for the instability parameter
was determined. The procedure applied in each test consisted
in, at first, to assign cross section dimensions for the members
of the frames assemblage and compute its horizontal stiffness
I
C1
, according to item 15.5.2 of ABNT [8] code (relation between
the wind load and the horizontal displacement at structure top).
After, the cross section of the walls assemblage was adjusted in
order to obtain the desired
I
C1
/
I
C
ratio.
The test proceeded with an initial second order analysis of the
frame-wall system, employing the
P
-
D
method. After, this second
order analysis was successively repeated, adjusting the values
of the vertical loads, until achieving the desired 10% increase on
the support global moment. Although an adjustment of the hori-
zontal stiffness would be more logical, the loads adjustment was
preferred because it made the 88 tests more agile to perform and
didn’t affect the results. The physical nonlinearity was considered
by means of the individual bar stiffness reductions expressed
by equations (10) and (11). The analysis was performed using
the same plane frame model of figure 3, with the sets of frames
and walls joined by hinges, since the formulation proposed in this
work is based just on that model. Later on, some cases were re-
analyzed by means of a method employing geometric stiffness
matrices, in order to confirm the results obtained by
P
-
D
method.
Table 5 – Values of
a
, varying the I /I ratio and the numbers of floors and spans
1
C1 C
Example:
1 2
3
4 5
6 7 8
I /I
C1 C
Floors: 5
5
10
10
20
20
30
30
1,00
0,515 0,514 0,528 0,519 0,569 0,534 0,608 0,591
0,95
0,552 0,557 0,567 0,563 0,605 0,590 0,639 0,635
0,90
0,572 0,584 0,594 0,592 0,630 0,621 0,656 0,656
0,85
0,590 0,603 0,613 0,614 0,650 0,644 0,675 0,676
0,80
0,610 0,619 0,629 0,632 0,663 0,662 0,690 0,690
0,70
0,626 0,641 0,653 0,657 0,687 0,687 0,710 0,711
0,60
0,641 0,656 0,671 0,676 0,702 0,702 0,724 0,727
0,50
0,652 0,665 0,685 0,689 0,716 0,716 0,734 0,735
0,40
0,662 0,671 0,695 0,699 0,724 0,726 0,743 0,740
0,20
0,675 0,681 0,714 0,715 0,738 0,736 0,753 0,752
0
0,683 0,683 0,726 0,726 0,749 0,749 0,764 0,764
5.2 Results discussion
The values of
a
1
obtained in the tests are listed in table 5. In or-
der to better interpret the results, it is appropriate to arrange the
eight examples according to the floors number and to consider two
ranges of the relative stiffness values:
I
C1
/
I
C
< 0,9 and
I
C1
/
I
C
> 0,9.
Figures 7 and 8 show, for each floors number, a graph represent-
ing the variation of the parameter a
1
found in the tests, as well as
the graph of
a
1
corresponding to formula (93).
On examining the results regarding to
I
C1
/
I
C
< 0,9, it is verified that
almost all the values for
a
1
obtained in the examples are below the
values predicted by formula (93) and listed in table 1. Thus, the
application of this formula results in upward errors, whose maxi-
mum values are mentioned in table 6. It can be clearly observed
that these errors decrease as the number of floors increases; they
are of 16,2% at 5 floors and drop to 3,1% at 30 floors, indicating a
trend to become null for a little more than 30 floors. This behavior
of equation (93) is much probably due to the adoption of figure 1-b
model, in place of figure 1-a one; in other words, the model with
an uniform and continuous distribution of floors and vertical loads
provides a reasonable accuracy only for buildings with more than
30 floors.
On the other hand, for values of
I
C1
/
I
C
larger than 0,9, indicative of
a high predominance of frames, the trend of decreasing errors with
the increase of floors also exists. However, in this case, it comes
along with another trend, consisting of downward errors generat-
ed by equation (93), increasing with the floors number, as can be
Table 6 – Maximum errors (%)
I /I
C1 C
5 floors
10 floors
20 floors
30 floors
≤
0,90
+16,2
+11,2
+6,0
+3,1
= 0,95
+10,7
+8,5
+3,6
–4,4
= 1,00
–1,2
–3,6
–10,5
–16,3