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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 6
Design of compression reinforcement in reinforced concrete membrane
(19)
-n
y
- n
xy
²
n
y
h ≤f
cd2
=0,6
(
1--f
ck
250
)
.f
cd
3.1.4 Case IV – Membrane without reinforcement
In order not to use reinforcement, there can be no tensile stresses
in membrane, thus loading conditions shall satisfy the following
inequations.
(20)
n
sx
=n
x
+|n
xy
|≤0
(21)
n
sy
=n
y
+|n
xy
|≤0
Therefore, in this case, it must only check if the compressive stress
in the concrete is less than the limit strength. For this verification,
differently from the other cases, it is used as a reference value f
cd1
of CEB [3] for the strength of concrete, because there is no crack-
ing in this case. From membrane equilibrium, this verification can
be written as:
(22)
(
- n
x
+n
y
2 + (
) n
x
-n
y
² 4 +n
xy
²
)
. 1 h
≤
f
cd1
=0,85.
(
1-
- -
f
ck
250
)
.f
cd
3.2 Considerations about the study
of compression reinforcement
Keeping the design cases proposed by CEB [3] in mind, it will be
examined which of them is consistent for studying compression
reinforcement.
Firstly, in case I, compression reinforcement could only be effec-
tively used if it were arranged in the direction of the cracks, which
would help reduce compressive stress in the concrete. However, in
this case, there is already an orthogonal grid of reinforcement, and
if another layer of bars is placed in the other direction, the solution
would be constructively bad, only recommended in special cases.
Another possibility would be to use a larger reinforcement area than
necessary to limit compression strain and to reduce stress in the princi-
pal compressive direction. However, this solution would lead to a brittle
rupture, because concrete would collapse before reinforcements yield.
For cases II and III, it is reasonable to think that placing reinforce-
ment in the direction that it was not reinforced, it will affect the
stress field in the membrane and it will reduce the compressive
stress in the concrete.
For case IV, for which there is no reinforcement, it is evident that if
reinforcement is placed appropriately in this membrane, it will help
to reduce the compressive stresses.
Therefore, this paper will only study design cases II, III and IV.
4. Strength model and verification
of compressive stress in concrete
In the method based on Baumann`s criteria presented, the strength
values
of concrete follow those recommended by the CEB [3]. How-
ever, this imposes a discontinuity of concrete strength between the
case in which the concrete is cracked and that in which it is intact.
Therefore, this work will be adopted a resistance model for the
concrete that optimizes the use of the material. To do so, the ob-
jective is to find strength values
for the concrete that are between
f
cd1
and f
cd2,
using the tensile strain that occurs perpendicular to the
compressive strain as a parameter.
Vecchio and Collins [2] propose a formulation which includes concrete
softening due to cracking differently from CEB [3]. For them, the loss of
strength of cracked concrete is related to principal tensile strain ε
1
im-
posed on the membrane. Equations 23 and 24 define the stress-strain
diagram for the compression in the concrete proposed by Vecchio and
Collins [2]. Considering the strain value that leads to stress peak in con-
crete ε’
c
=2‰, equation 24 is obtained. Figure 2 shows this model.
(23)
f
c2
=f
c2max
.
[
2.
(
(
ε
2
ε`
c
)
)
-- -ε
2
ε`
c 2
]
(24)
f
c2max
= f`
c
0,8+170. ε
1
≤
f`
c
Expression 23 is similar to that suggested by CEB [3] for the stress-
strain diagram of concrete, only changing the strength limit. In this
paper, it will use the limits proposed by CEB [3], but interpolated by
equation 24. Thus, it follows that:
(25)
σ
c
=f
c2max
.
[
2. ( ε
2
ε`
c
) -- -( ε
2
ε`
c
)
2
]
Figure 2 – Stress-strain diagram for cracked
concrete in compression (Vecchio e Collins, 1986)