Basic HTML Version
827
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 6
T. F. SILVA | J. C. DELLA BELLA
(39)
ε
1
+ε
2
=ε
x
+ε
y
Then:
(40)
ε
1
=ε
yd
-ε'
c
Also through the Mohr circle, by replacing θ with θ*, it follows that:
(41)
ε
x
= ε
1
+ε
2
2 +
(
ε
1
-ε
2
2
)
.cos2θ
*
Replacing 40 in 41, θ* is given by equation 42:
(42)
θ
*
= arccos
(
ε
yd
ε
yd
-2.ε
' c
)
2
Table 1 shows the values
of θ
1
, θ
2
and θ* to steels determined by
NBR 6118 [14]. It can be observed that for steels CA-50 and CA-60
there are no values of
θ
1
. This is because for strain values
assumed
by hypothesis for this problem, the strength of the concrete never
reaches the value of f
cd1
for these steels. Thus, the strength of con-
crete reaches its maximum when θ = 0º. Thus, it equalizing strength
with stress in concrete, if θ1 exists and θ = 0, then nxy = 0 and:
(43)
f
cd1
= n
c
h
If θ
1
exists and 0 < θ ≤ |θ
1
|, then:
(44)
f
cd1
= 2.n
xy
h.sen(2θ)
Table 1 – Values of
θ
1,
θ
2 and
θ
* for steels
prescribed by NBR 6118 to case III
(‰)
yd
o
|
θ
| ( )
1
o
|
θ
| ( )
2
o
|
θ
*| ( )
CA-25
1,04
12,17
42,74
39,07
CA-50
2,07
DOES NOT EXIST 31,74
35,03
CA-60
2,48
DOES NOT EXIST 26,79
33,74
If θ
1
does not exist and θ = 0, so n
xy
= 0 and:
(45)
f
cd1
0,8+170.
[
2.ε
yd
-ε'
c
.(1-cos2θ)
(1+cos2θ)
]
= n
c
h
If θ
1
exists and |θ
1
|< θ < |θ
2
| or if θ1 does not exist and 0 < θ < |θ
2
|,
then:
(46)
f
cd1
0,8+170.
[
2.ε
yd
-ε'
c
.(1-cos2θ)
(1+cos2θ)
]
= 2.n
xy
h.sen(2θ)
If |θ
2
| ≤ θ < |θ*|, then:
(47)
f
cd2
= 2.n
xy
h.sen(2θ)
Considering this situation, the objective is to find for which values
of forces it is possible to design compression reinforcement. For
normal forces, there is no mathematical limit, there is only con-
structive limit to the reinforcement ratio prescribed by NBR 6118
[14]. For shear force, there is a limit, but the formulation of which
varies with the kind of steel adopted. This is because limits
θ
1
,
θ
2
and
θ
* are different for each steel. For CA-25, as |
θ
*| <|
θ
2
|, equation
47 will never be valid. Thus, it should be known which maximum
value of n
xy
can be assumed for this steel. Then, taking back equa-
tion 46, it is possible to demonstrate that:
(48)
|n
xy
|≤ f
cd1
.h 2 .
(
(
sen(2.|θ
xy
|)
0,8+170.
[
2.ε
yd
-ε
' c
.(1-cos (2.|θ
xy
|))
(1+cos? (2.|θ
xy
|))
]
In which:
(49)
|θ
xy
|=33,76°
For CA-50 and CA-60, as |θ*| > |θ
2
|, equation 47 is valid. From it, it
is possible to demonstrate that:
(50)
|n
xy
|≤ f
cd2
.h.sen(2|θ
*
|)
2
Therefore, if n
xy
respects the condition imposed by 48 or 50, the
problem has a solution, in other words, there is a reinforcement
which will decreases stress in concrete until its maximum strength.
Table 2 shows the maximum values
of θ for each kind of steel.