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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 3
M. C. MARIN | M. K. EL DEBS
Because the present study did not consider the effect of such axial
force variations, the M x N x 1/r diagrams were constructed and
used in the finite element model in one step.
Initially, the stiffness of elements in a precast concrete building
with 6 floors was analyzed. According to the above methodology,
the reference bending moment for evaluating the stiffness reduc-
tion in the elements depends on the resistant moment in the sec-
tion and not on the loads acting on the element.
For each section of the elements in the study, M x N x 1/r diagrams
were constructed and analyzed, and the stiffness reduction coef-
ficients were found. Functions to reduction stiffness related to the
stiffness reduction coefficient and the dimensionless axial force
are proposed. The reduction coefficients found using the stiffness
functions recommended by different standards are compared with
the values found from the M x N x 1/r diagrams.
After defining the value of the axial force and the load combinations
acting on a section of a column, the M x N x 1/r diagrams were calcu-
lated. Figure 5 shows an example of a diagram using a stress value
(on the concrete) of 1.1 f
cd
and increasing the acting loads by
γ
f
(N
d
).
Table 6 shows the coefficients obtained from the M x N x 1/r di-
agram that used a stress value (on the concrete) of 1.1 f
cd
and
increased the acting loads by
γ
f
/
γ
f3
(N
d
/
γ
f3
). This enabled the de-
termination of the stiffness reduction coefficients corresponding to
secant stiffness.
Analyzing the stiffness reduction coefficients shown in Table 6
with respect to the calculation combinations in ULS, the central
column’s stiffness reduction coefficient varied from approximately
0.35 to 0.6, and the lateral column’s coefficient varied from ap-
proximately 0.35 to 0.5. Because of the greater effect of the axial
force in the third combination, the stiffness reduction coefficient
found in the third combination of loads is greater than the stiffness
reduction coefficients found in the first and second combination.
Table 6 – Stiffness reduction coefficient in columns (P50x50) for structure
2
with modulation 7.5m and live load 3 kN/m
st
a
: Stiffness reduction coefficient in columns for 1 load combination (ULS);
.1
nd
a
: Stiffness reduction coefficient in columns for 2 load combination (ULS);
.2
rd
a
: Stiffness reduction coefficient in columns for 3 load combination (ULS).
.3
a
.1
a
.2
a
.3
FLOOR
CC
LC
CC
LC
CC
LC
6
5
4
3
2
1
0.366
0.343
0.345
0.333
0.375
0.347
0.430
0.397
0.384
0.366
0.446
0.404
0.493
0.441
0.422
0.409
0.501
0.453
0.514
0.492
0.459
0.443
0.527
0.498
0.539
0.509
0.495
0.483
0.566
0.516
0.583 0.528
0.507 0.501
0.625 0.538
Table 7 – Stiffness reduction coefficient calculated in each recommendation
n
: Dimensionless axial force;
l
: Slenderness ratio.
n
l
M x N x 1/r
NBR 6118:2003
ACI 318-08
FIB
NBR
9062:1985
PCI
0.58
27.71
0.625
0.800
0.471
0.429
0.758
0.162
0.58
63.74
0.625
0.800
0.471
0.498
0.758
0.038
0.03
27.71
0.333
0.500
0.471
0.280
0.758
0.015
0.03
63.74
0.333
0.500
0.471
0.283
0.758
0.038
Figure 5 – M x N x 1/r diagram for first
load combination in central column P50x50