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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 3
M. C. MARIN | M. K. EL DEBS
The GNL was analyzed according to the complete Newton-Raph-
son method. Displacement control was the criterion used to deter-
mine when the iterative process was stopped. The defined toler-
ance was 0.5%, meaning that the iterative process was interrupted
when there the increase in displacement relative to the previous
iteration was no more than 0.5%. The non-linear analysis was per-
formed by dividing the load into 10 steps.
According to NBR 6118:2003 [1], non-linearity can generally be
considered through each section’s bending moment x axial force
x curvature relationships (M x N x 1/r diagrams), where the re-
inforcement and acting axial force are supposedly known. NBR
6118:2003 [1] describes two ways of using the M x 1/r diagram; the
first is designed for the ultimate limit state, and the second is de-
signed for evaluating the secant stiffness of the elements. Figure 1
shows the M x 1/r diagram from NBR 6118:2003 [1].
According to NBR 6118:2003 [1], to calculate ULS, the design
value of the compressive strength of concrete should be multiplied
by 0.85. As explained by CARVALHO and FIGUEIREDO [15], be-
cause the concrete shows greater compressive strength in short-
term trials, the value of 0.85 f
cd
is assigned to the duration of the
compressive strength tests. In typical structures, the load contin-
ues to act on the structure throughout its entire useful life. Under
dead load, concrete’s compressive strength decreases over time,
a phenomenon called the Rüsch effect.
According to FRANÇA [16], calculating the stiffness from constitu-
tive relationships based on design values of the concrete strength
can lead to overestimating the effects of non-linearity. To account
for stiffness, the design value of the concrete compressive strength
should be multiplied by 1.10. This coefficient considers the fact
(19)
θ = 27 kL/r - 0.05
and
P
represents the axial load in the column in a first-order analysis;
0
P
represents the maximum acceptable centered load on the col-
umn;
κ
represents the coefficient of effective length of the column con-
sidering the boundary conditions;
L
represents the length of the column;
r
represents the radius of gyration of the cross section.
In this case, it is noteworthy that the expression takes into ac-
count the geometric characteristics and connections in the struc-
tural elements.
As shown by Table 1, a comparative table of the factors considered
in each recommendation, a) there is a large difference between the
factors considered and b) NBR 6118:2003 [1] is the only scenario
with fixed values. It is important to note that the factors considered
are only applied to columns.
3. Models for analysis
The precast concrete structure considered in this study was modeled
as a plane frame, using ANSYS
®
software [9] for structural analyses
with finite elements. The non-simplified consideration of PNL implies
knowing the M x N x 1/r diagram (in other words, the stiffness) for
each section in which there is a change in stress, cross-section, rein-
forcement, concrete cover, and
strength of concrete. Thus, structures
that are discretized into a greater number of finite elements have more
representative solutions. Using the ANSYS
®
software [9], the constitu-
tive relationship between beams and columns can be represented by
the M x N x 1/r diagram using the beam element BEAM188.
The discretization adopted for modeling the structure through the
finite element method employed 8 finite elements for each column
section, where each section corresponds to the region between
floors. With regard to the beams, 16 finite elements were used for
each beam section, where each section was defined by the region
between corbels. The corbels were discretized into a finite ele-
ment, and the stiffness of the corbels was defined by the product
E
ci
I
c
. The elements of connection were modeled using COMBIN39,
permitting the bending moment x rotation relationship to be repre-
sented in a non-linear and asymmetric way.
Table 1 – Factors considered in each recommendation
VALUES
AXIAL
FORCE
CREEP
REINFORCEMENT
SLENDERNESS
NBR 6118:2003
FIXED
NO
NO
NO
NO
ACI 318-08
VARIABLE
NO
YES
YES
NO
FIB (2002)
VARIABLE
YES
YES
YES
YES
NBR 9062:1985
VARIABLE
NO
NO
YES
NO
PCI (1988)
VARIABLE
YES
YES
NO
YES
Figure 1 – Bending moment x curvature
[NBR 6118:2003]