Basic HTML Version
119
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
R. J. ELLWANGER
observed in table 6 for
I
C1
/
I
C
= 0,95 and 1,00. This trend is due to
equation (93) taking into account that the frames distort only due to
global shear. Smith and Coull [13] state that, in slender buildings, the
frames global bending, due to the columns axial deformation, can
contribute significantly to the horizontal displacements. It is about
the same distortion pattern of walls, making the limit coefficient
a
1
to increase. Thus, as much higher the building and the greater pres-
ence of frames, greater will be the errors generated by a formulation
that neglects this effect. In the case of bracing systems composed
exclusively by frames (
I
C1
/
I
C
= 1), table 6 shows maximum errors
varying between –1,2% (5 floors) and –16,3% (30 floors).
Still regarding to the case
I
C1
/
I
C
= 1, the values of
a
1
mentioned
in the first line of table 5 hint that the limit
a
1
= 0,5, prescribed by
ABNT [8] for bracing systems composed exclusively by frames, is
conservative, especially in buildings with more than 10 floors. On
the other hand, the values of a
1
found for
I
C1
/
I
C
= 0, mentioned in
the last line of table 5, indicate that the limit
a
1
= 0,7, prescribed for
bracing systems composed exclusively by walls/cores, is also con-
servative for buildings with more than 10 floors. However, in build-
ings with less than 10 floors, the contrary can occur; values slightly
lower than 0,7 were found in buildings with 5 floors. Furthermore,
the adoption of the fixed value
a
1
= 0,6 for wall-frame and core-
frame bracing systems should be conditioned to a minimum limit of
the walls contribution for the bracing stiffness, especially in lower
buildings. Interpolations performed in the values of table 5 indicate
that the proportion of the walls gross inertia in relation to the total
one should be at least 18% in example 1, 14% in example 2 and
12% in examples 3 and 4.
6. Conclusions
The limit values
a
1
for the instability parameter, obtained in the
examples of this work and mentioned in table 5, vary from a mini-
mum of 0,514 in example 2 until a maximum of 0,764 in examples
7 and 8. The proportion between these extreme values is close to
1,5:1. Since their computation includes a square root extraction,
the proportion between the radicands (vertical load/horizontal stiff-
ness relations) associated to these extreme values is more than
2:1. The extent of this variability shows the importance of having
a way of predicting a limit
a
1
appropriated to the relation
I
C1
/
I
C
and
the floors number of a given building to be designed, in place of the
fixed values prescribed by ABNT [8].
Equation (93) represents an initial attempt to accomplish such pre-
diction. The relatively good accuracy attained in examples 7 and
8 for
I
C1
/
I
C
< 0,9 denotes that this aim is possible to be achieved
and efforts deserve to be done in order to carry it out. In order to
remedy the errors occurred in the other cases (predominance of
frames and low number of floors), the effect of the columns axial
flexibility has to be introduced into equations (16) and (50), mak-
ing the curve of
a
1
, corresponding to equation (93) and depicted
in figures 7 and 8, decline not so much in its final segment; it has
also to be searched a way to adjust the formulation to the variation
of the floors number. Another subject to investigate is the viability
of including the variability of physical nonlinearity influence in the
frames horizontal stiffness (actually, this influence is considered
through a constant factor).
It must be emphasized that all of this has to be done in such a man-
ner to keep the formulation simplicity, just one of the greater advan-
tages of the instability parameter utilization. Finally, it must be ac-
centuated the need of adopting a more realistic analysis model for
the tests: modeling of the structure as a three-dimensional frame,
considering the floors as rigid diaphragms; variation of the wind
load along the building height; effectuation of the nonlinear analy-
sis through an incremental-iterative method; and a more accurate
consideration of physical nonlinearity, for example, by means of
moment-curvature relations.
7. Acknowledgment
I would like to thank Prof. Eng. Mário Franco, for providing pre-
cious material for this work achievement.
8. References
[01] Beck, H. and König, G.; Haltekräfte im Skeletbau.
In
: Beton- und Stahlbetonbau, n. 62, tome 1
(pp. 7-15) and tome 2 (pp. 37-42), Berlin, 1967.
[02] CEB – Comité Euro-International du Béton; CEB/FIP
Manual of Buckling and Instability, The Construction
Press, Lancaster, 1978.
[03] Vasconcelos, A.C.; Origem dos Parâmetros de
Estabilidade
a
e
g
z
.
In
: Revista IBRACON de
Estruturas, n. 20, pp. 56-60, São Paulo, 1998.
[04] Sussekind, J.C.; Curso de Concreto, vol. 2, Porto
Alegre, Ed. Globo, 1984.
[05] Franco, M.; Problemas de Estabilidade nos Edifícios
de Concreto Armado.
In
Colóquio IBRACON sobre
Estabilidade Global das Estruturas de Concreto
Armado, São Paulo, 1985.
[06] Franco, M.; Global and Local Instability of Concrete
Tall Buildings,
In
: International Symposium for Shell
and Spatial Structures, Proceedings, vol. 2,
pp. 1327-36, Milan, 1995.
[07] Franco, M. and Vasconcelos, A.C.; Practical
Assessment of Second Order Effects in Tall Buildings.
In
: Colóquio do CEB-FIP Model Code 1990,
pp. 307-24, Rio de Janeiro, 1991.
[08] ABNT – Associação Brasileira de Normas Técnicas;
NBR 6118 – Projeto de Estruturas de Concreto –
Procedimento, Rio de Janeiro, 2007.
[09] Stamato, M.C.; Associação Contínua de Painéis de
Contraventamento (Publicação 157), São Carlos,
EESC/USP, 1972.
[10] Kantorovich, L.V. and Krylov, V.I.; Approximate
Methods of Higher Analysis, New York, Interscience
Publishers Inc., 1964.
[11] Pinto, R.S. and Ramalho, M.A.; Inércia equivalente
das estruturas de contraventamento de edifícios em
concreto armado.
In
: Cadernos de Engenharia de
Estruturas, São Carlos, v. 9, n. 38, p. 107-136, 2007.
[12] Schueler, W.; High-rise Building Structures, New
York, John Wiley & Sons, 1977.
[13] Stafford Smith, B. and Coull, A.; Tall Building
Structures: Analysis and Design, New York, John
Wiley & Sons Inc., 1991.