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550
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 4
Design of slender reinforced concrete rectangular columns subjected to eccentric loads
by approximate methods
have longitudinal and transversal (stirrups) reinforcement, and
were tested in one axis bending with axial compressive loads. Col-
umns fabricated with concrete containing light aggregates or fibers
are not included.
The most important characteristics of the columns in the databank
are shown in Tables 1 and 2. Each table corresponds to a range of
concrete compressive strength: up to 55 MPa and above 55 MPa.
Each table also presents the number of columns tested by each
research team, their respective geometrical characteristics, the
longitudinal reinforcement ratio as well as the concrete compres-
sive strength. The indicated concrete compressive strength were
obtained from 15 cm x 30 cm cylinder specimens.
2.3 Comparative Study
The comparative study between the actual and predicted second
order effects can be quantified based on the ratio of the test failure
moment
M test
to the predicted one. The test failure moment
M test
provides a close estimate of the true capacity. It can be determined
from the loads and the total lateral displacements measured at the
column failure. The design moment
M pred
is calculated using
equations 4 and 7 respectively. For the evaluation of the predicted
moment
M pred
, all material resistance factors were set equal to
one. Further, the measured concrete compressive strength
c
f
of
each test specimen was used in determining
M pred
.
For each column the ratio
M test
/
M pred
was calculated. Statistical
analyses of this ratio include its average
m,
the median
m d ,
the
standard deviation
SD
, the coefficient of variation
CV
as well as the
maximum and minimum values. The average of
M test
/
M pred
is
used as a measure of the conservative bias of the procedure while
the coefficient of variation is taken as an indication of accuracy.
With the objective of evaluating the reliability and of comparing the
performance of shear design code equations for reinforced con-
crete beams, COLLINS [4] developed a demerit point scale meth-
odology. Considering safety, precision and economy, a score is
attributed for each range of
M test
/
M pred
ratio: these values are
The symbols used are also explained in the notation.
The dimensionless stiffness
κ
is needed to calculate
M d,tot
and is
also a function of
M d,tot
. Thus an iterative process must be used.
However, Scadelai [2] has shown that an iterative process is not
necessary and
M d,tot
can be calculated from:
(7)
(
)
A,d1
2
1
2 d
1
2
2
2 1
tot ,d
M
10
M.25 k N.h.2M.10 k k M.5
M
³
+ -
+ + -
=
with
(8)
M
Ad,1
b
1
M
a=
(9)
3840
1
2
1
l
-=
k
(10)
d
1
2
Nhk k
=
2.2 Slender reinforced concrete columns database
The database for this analysis was originally assembled by Souza
[3] as part of his final undergraduate report. It consists of slender
columns tested up to failure in laboratories worldwide. All columns
Table 1 – Column database (f
≤
55 MPa)
c
Research team
Number
of tested
columns
b
(cm)
h
(cm)
l
e
(cm)
l
f
c
(MPa)
r
l
(%)
ENCISO [5]
4
25
15
312
72
46,9 to 53,6 1,30 to 4,30
ADORNO [6]
6
25
12
215
62 36,9 to 42,5
1,05
LEE and SON [7]
6
21
12
138 - 210
40 - 61
34,9 to 41,8
1,13
DANTAS [8]
5
25
12
300
86,7
34 to 38
1,57
LIMA JÚNIOR [9]
3
15
15
195
45
39,2
2,18
SANTOS [10]
11
25
12
200 - 250
58 - 72
37,8 to 45,8
1,57
MELO [11]
4
25
12
300
86,7
39,6
1,57
GALANO and VIGNOLI [12]
6
10
10
212
73,4
43,1
2,01 to 4,52
CLEASON and GYLLTOFT [13] 6
12 - 20
12 - 20
260 - 420
55 - 75
33 to 43
2,1 to 3,2