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1. Introduction
Reinforced concrete (RC) columns are linear structural elements,
usually cast vertically, responsible for carrying the loads from floors
to the foundations. The stability of any given structure is directly
linked to the stiffness and strength of the columns. Thus, the de-
sign of reinforced concrete columns must include local checks as
well as global analysis of the structural system.
Concrete codes worldwide require that columns must be designed
to resist not only axial and bending moments computed from an or-
dinary first order frame analysis including allowances for construc-
tion imperfections but also moments due to internal force effects
resulting from deflections (second-order effects). Thus the design
of concrete columns must be based on the factored forces and
moments from a second-order analysis considering material non-
linearity and cracking, as well as the effects of member curvature
and lateral drift, duration of the loads, shrinkage and creep, and
interaction with the supporting foundation. Column sections, under
the above conditions, must then be designed to ensure that there
is neither instability nor material failures.
Use of such an analysis to determine column compressive axial
forces and bending moments for section design is the most ra-
tional approach. But this analysis is very complex. Thus concrete
codes worldwide allow the use of approximate design methods for
slender columns. The simplified methods shown in NBR 6118 [1]
are
approximate curvature method and approximate stiffness pro-
cedure
. Due to the responsibility of columns in the stability and
strength of concrete structures, these approximate approaches
must provide adequate safety in the design.
In this scenario, the goal of this paper is to analyze NBR 6118
[1] approximate design criteria for slender rectangular columns
subjected to eccentric loads with respect to safety, precision and
economy, by comparing code based calculations with respect to
experimental results of columns built with conventional concrete
(
f
c
≤ 55 MPa). The investigation also includes columns with con-
crete having compressive strength above 55 MPa since NBR 6118
procedures are being changed to allow the use of concrete with
compressive strength above 55 MPa.
2. Methodology
2.1 NBR 6118 approximate design approaches
for slender columns
The NBR 6118 [1] provides two simplified design procedures for
the evaluation of second order effects in slender columns: approxi-
mate curvature method and approximate stiffness procedure. The
simplifications in both procedures are related to geometric and ma-
terial nonlinearities.
2.1.1 Approximate curvature method
The approximate curvature method is applicable to RC columns
with slenderness ratios less or equal to 90 and symmetrical lo-
cation of the reinforcement. It can only be used in columns sub-
jected to axial loads and bending on one axis. The geometrically
nonlinear behavior is simplified by assuming a deformed shape
represented by a sine curve. The material nonlinearity is taken into
account by an approximate equation for the curvature at the critical
column cross-section.
The lateral displacement
2
e
due to the second order effects is
given by:
(1)
r
e
e
1
10
2
2
×
=
l
The curvature is calculated from:
(2)
h
vh r
005 ,0
)5,0 (
005 ,0 1
£
+
=÷
ø
ö
ç
è
æ
with
(3)
5,0
.
³
=
dc
d
f A
v
C
N
The column maximum bending moment is then equal to:
(4)
(
)
h03,05,1N M with e.N M
M
d A,d1 b
2 d A,d1 b
tot ,d
+
³
+
=
a
a
These symbols used are explained in the notation.
2.1.2 Approximate stiffness procedure
The approximate stiffness procedure is applicable to RC columns
subjected to combined flexure and axial loads with unsupported
length to radius of gyration ratio less or equal to 90 and symmetrical
location of the reinforcement. It can only be used in columns with
rectangular cross-sections. The deformed shape in this case is also
represented by a sine curve. The material nonlinearity is taken into
account by an approximate equation for column flexural stiffness.
The maximum design bending moment in a column is equal to:
(5)
Ad
Ad
b
tot d
M
k
M
M
,1
2
,1
,
120
1
.
³
-
=
n
l
a
where the dimensionless stiffness
k
is calculated from:
(6)
÷÷
ø
ö
çç
è
æ
+
=
d
tot d
Nh
M
k
.
.51 .32
,
n
549
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 4
J. M. CALIXTO | T. H. SOUZA | E. V. MAIA