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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 6
T. F. SILVA | J. C. DELLA BELLA
same steps used in case III, but using the formulation found here-
in. First, if n
xy
= 0, then θ = 90 °. If θ
1
exists and |θ
1
| ≤ θ <90 °, then:
(60)
θ= arcsen
(
2.n
xy
f
cd1
.h
)
2
If θ
1
exists and |θ
2
| < θ < |θ
1
| or if θ
1
does not exist and |θ
2
| < θ
< 90º, then
:
(61)
θ= arcsen
(
(
2.n
xy
f
cd1
.h
0,8+170.
[
[
2.ε
yd
-ε
' c
.(1+cos2θ)
(1-cos2θ)
2
Finally, if |θ*| < θ ≤ |θ
2
|, then:
(62)
θ= arcsen
(
2.n
xy
f
cd2
.h
)
2
For case II, ε
yd
= ε
y
. So, with ε
2
= ε’
c
e θ, it is possible to find the
value of ε
1
and then ε
x
. Thus, it follows that:
(63)
ε
x
= 2.ε
yd
-ε
' c
.(1+cos2θ)
(1-cos2θ)
+ε'
c
-ε
yd
It is possible to calculate the forces in the reinforcement using
equations 64 and 65. The reinforcements are given by:
(64)
a
sx
= n
sx
σ
x
= n
sx
E.ε
x
(65)
a
sy
= n
sy
σ
y
= n
sy
E.ε
y
= n
sy
E.ε
yd
= n
sy
f
yd
6. Design of compression reinforcement
to case IV
Case IV is different from cases II and III because the concrete
strength conditioning is not the same. As in this case the membrane
is in biaxial compression state, the concrete strength could be even
higher than the value of f
cd1
, as recommended by the CEB [3]. How-
ever, in this study, concrete strength will be considered equal to f
cd1
.
The objective of the formulation that will be presented is to design
the reinforcement in x and y directions for membranes in which
stress in concrete is higher than its strength.
First, it is assumed that stress in concrete is equal to its limit in
ULS. Another problem hypothesis is that the membrane is always
in biaxial compression state; in other words, the inclusion of rein-
forcement which leads to tensile stress in membrane will not be
contemplated by this study. Thus, the hypotheses are:
1. The tensile strength of concrete is null
2. The bond between reinforcement and concrete is perfect
3. The membrane is always in biaxial compression state
4. The directions of principal strains and the directions of principal
stresses coincide
5. The concrete strength is given by f
cd1
6. The principal compressive strain is equal to the strain resulting
in the peak stress in concrete (ε
2
= ε’
c
). Thus, the principal com-
pressive stress is equal to the compressive strength (σ
c
= f
cd1
).
7. The effect due to aggregate interlock will not be considered
8. The tension-stiffening effect will not be considered
By equilibrium of membrane, the following expressions are obtained
where n’
c
is the force in the direction of minimum compression.
(66)
n
c
=-n
x
+n
xy
.cotgθ+n
sx
(67)
n
c
=n
sy
-n
y
+n
xy
.tgθ
(68)
n
c
=n
c '
+n
xy
.(tgθ+cotgθ)
(69)
n
c
=- (n
x
-n
sx
)+(n
y
-n
sy
)
2
+ ((n
x
-n
sx
)-(n
y
-n
sy
))
2
4
+n
xy
²
(70)
n
c '
=- (n
x
-n
sx
)+(n
y
-n
sy
)
2
- ((n
x
-n
sx
)-(n
y
-n
sy
))
2
4
+n
xy
²
6.1 Design limits
The objective in this item is to define the cases for which it is possi-
ble to design compression reinforcement. By hypothesis, the mem-
brane is always in biaxial compression state, then n’
c
≥ 0. Thus,
from equation 68, it follows that:
(71)
sen2θ≥ 2.n
xy
n
c