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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 6
Design of compression reinforcement in reinforced concrete membrane
For case II, equation 34 modifies and results in expression 35:
(35)
ε
2
=ε`
c
. 1- 1-
σ
c
f
cd1
0,8+170.
[
2.ε
yd
-ε
2
.(1+cos2θ)
(1-cos2θ)
]
Equations 34 and 35 can be solved by iterative methods. Assum-
ing an initial value to
ε
2
for which the function exists (i.e., the radi-
cand will not be negative), it will converge to the solution. If the
radicand assumes a negative value in any iteration, then the prob-
lem has no solution, and therefore the stress in concrete is higher
than the maximum limit.
5. Design of compression reinforcement
for cases II and III
All the demonstrations in this item will be made
only for design case
III. Case II is analogous and only its final formulation will be presented.
For those cases in which the compressive stress in concrete is
higher than the strength calculated as shown in item 4, it should
be checked if adopting compressed reinforcement in the direction
in which there was no reinforcement previously, it will be effective
to decreases the stress in concrete so that it will be lower or equal
to the strength. First, for the purposes of this problem the same
assumptions given by the method based on Baumann`s criteria,
presented in item 3, will be used.
Moreover, some considerations about strain must be made. First,
it will be admitted that the strain in the x direction is equal to the
yield stress in steel. This assumption limits the strain in the mem-
brane, optimizing the compressive strength of the concrete, be-
sides resulting in a reinforcement area in which ductile rupture in
ULS occurs. In other words, even if to solve the problem it would
be necessary to over-reinforce the membrane, this result will be
discarded because the membrane would collapse in a brittle way.
Furthermore, it is assumed that strain
ε
2
is always equal to
ε
’
c
, lead-
ing the concrete to the strength limit and, consequently, reducing
the consumption of reinforcement. Summarizing the hypotheses,
it follows that:
1. The cracks are approximately parallel and straight.
2. The tensile strength of concrete is null
3. The dowel effect will not be considered
4. The effect due to aggregate interlock will not be considered
5. The bond between reinforcement and concrete is perfect
6. The tension-stiffening effect will not be considered
7. The directions of principal strains and the directions of principal
stresses coincide
8. The strain in x direction is equal to the yield strain of steel (ε
x
= ε
yd
).
9. The principal compressive strain is equal to the strain resulting
in the peak stress in concrete (ε
2
= ε’
c
).
5.1 Design limits
With these hypotheses, it is intended to determine the cases in
which it is possible to design compression reinforcement. Thus,
firstly, a membrane subjected to stresses such that tensile rein-
forcement in the y direction is not necessary, therefore, it lies in de-
sign case III, and the compressive stress in the concrete is higher
than the strength f
c2max
, as shown in item 4 is assumed. As hypo-
thetically
ε
x
=
ε
yd
and
ε
2
=
ε
’
c
, the compressive strength of concrete
is given by equation 36.
(36)
f
c2max
=
f
cd1
0,8+170.
[
2.ε
yd
-ε'
c
.(1-cos2θ)
(1+cos2θ)
]
In which:
f
cd2
≤f
c2max
≤f
cd1
The graph that describes the strength as a function of
θ
is shown
in Figure 4. For case III, all the functions of
θ
have domain 0 ≤
θ
≤
|45º|. For case II, the domain is |45º| ≤
θ
≤ |90º|.
It can determine the values
of θ
1
and θ
2
shown in Figure 4. Angle θ
1
is
the one which equates f
c2max
at f
cd1
. Thus, it can be demonstrated that:
(37)
θ
1
= arccos
(
0,00118-2.ε
yd
+ε
' c
(ε
' c
-0,00118)
)
2
As the cosine function produces the same result, no matter the
angle signal, both positive and negative θ
1
are solutions. Similarly,
θ
2
is the value that equates f
c2max
at f
cd2
. Thus, it follows that:
(38)
(
)
θ
2
= arccos 0,003627-2.ε
yd
+ε
' c
(ε
' c
-0,003627)
2
Also for θ
2
, both positive and negative solutions satisfy equation 38.
However, if θ exceeds a certain limit, strain ε
y
assumes positive
values
.
Thus, the area of reinforcement in y results in negative
values, which is not physically possible. As by hypothesis ε
x
= ε
yd
and ε
2
= ε’
c
, it can calculate to what values
of θ, ε
y
is lower than
0. The objective is to find θ* for which ε
y
= 0. From Mohr circle, it
follows that:
Figure 4 – Compressive strength of concrete
as a function of
θ
to case III