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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 6
T. F. SILVA | J. C. DELLA BELLA
(26)
f
cd2
≤
f
c2max
= f
cd1
0,8+170. ε
1
≤
f
cd1
The maximum strain used in this work is the same as that suggested
by CEB [3] for zones subjected to axial compression, and this limit is 2
‰. Thus, Figure 3 shows the stress-strain diagram used in this work.
4.1 Verification of compressive strength of concrete
The strength limit for concrete is herein calculated considering the con-
cepts presented in item 4. Thus, the way to check if the compressive
force respects this limit is different from that presented in the method
based on Baumann`s criteria, because the concrete capacity now de-
pends on the tensile strain to which the membrane is subjected in ULS.
In case IV, verification of concrete is the same of that in item 3,
because in this case there is no tension in the membrane and the
compressive strength of concrete is always given by f
cd1
.
For the cases II and III, it should be first checked if:
(27)
σ
c
= n
c
h ≥f
cd1
This study admits that f
cd1
is the maximum limit for the compressive
strength of concrete in any case. If inequation 27 is satisfied, the
compressive stress in the concrete is above the limit and, it should
thus evaluate the possibility of using compression reinforcement.
The way to do this evaluation will be presented in item 5.
For the case in which inequation 27 is not satisfied, it should be
verified if:
(28)
σ
c
= n
c
h
≤
f
cd2
As f
cd2
is the lowest limit for the compressive strength of concrete,
if expression 28 is satisfied, the compressive stress in the concrete
respects the strength limit imposed and it will not therefore be nec-
essary to use compression reinforcement. If inequations 27 and 28
are not satisfied, it consequently follows that:
(29)
f
cd2
≤
n
c
h
≤
f
cd1
In this case, the strains in the membrane must be considered to de-
termine the strength limits to be used, because it will depend on
ε
1
.
4.1.1 Calculation to determinate the limit of compressive
strength of the concrete
The objective of this item is to find the value of f
cd2max
. However, it
depends on the strain of the membrane. A calculation method based
on that presented by Jazra [12] will be presented. This calculation is
valid for cases II and III. Due to their being analog, changing just the
reinforcement position (y axis to case II and x axis to case III), only
case III will be described. For case II, equation 30 must be replaced
by the equivalent equation to
ε
y
and the same process must be re-
peated. Thus, from Mohr circle, it follows that:
(30)
ε
x
= ε
1
+ε
2
2 +
(
ε
1
-ε
2
2
)
.cos2θ
So:
(31)
ε
1
= 2.ε
x
-ε
2
.(1-cos2θ)
(1+cos2θ)
Therefore:
(32)
f
c2max
=
f
cd1
0,8+170.
[
2.ε
x
-ε
2
.(1-cos2θ)
(1+cos2θ)
]
From equation 25 it is possible to express the compressive strain
as a function of strength.
(33)
ε
2
=ε`
c
. 1- 1- f f
c2max
)
)
Considering by hypothesis that
ε
x
is equal to the yield strain of
steel, it follows that:
(34)
ε
2
=ε`
c
. 1- 1-
σ
c
f
cd1
0,8+170. 2.ε
yd
-ε
2
.(1-cos2θ)
(1+cos2θ)
Figure 3 – Stress-strain diagram adopted