Basic HTML Version
578
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 5
Shear force and torsion in reinforced concrete beam elements: theoretical analysis based
on Brazilian Standard Code ABNT NBR 6118:2007
on the design value of shear force (V
Rd3
) and the design shear
resistance (V
Rd2
). It differs from model I, in which the value of V
c
is constant. The value of V
c
equals V
c1
in flexure and flexotraction,
in which the neutral axis crosses the section of the structural ele-
ment. This value can be obtained through equation (6).
(6)
0c
0c
2Rd
sd
2Rd
1c
V
V V
V V
V
The previous expression was derived by linear interpolation, and
the value of V
c1
equals V
c0
when the value of V
Sd
equals V
c0
. Si-
milarly, V
c1
equals zero when the design value of shear force V
Sd
equals the value of the design shear resistance (V
Rd2
).
The design shear resistance in relation to the compression strut
capacity is calculated through equation (7), and the calculation of
the transverse reinforcement is based on equation (8). It is obser-
ved that when angle θ is 45° in equation (7), this equation equals
the design shear resistance equation obtained using calculation
model I.
(7)
gcot
g cot
sen d b f) 250 / f1( 54,0 V V
2
w cd
ck
2Rd
sd
(8)
sen g cot
gcot
fd9,0
s
A
V
ywd
sw
sw
Considering the offset (a
ℓ
) of the force in the tensile chords diagram
for calculation model II, it depends on the value of the effective
height (“d”) of the structural element, the angle θ of inclination of
the strut and the angle a of inclination of the stirrups. The value of
a
ℓ
must be limited to the value of the effective height of the element.
Thus, the less the value of angle θ of strut inclination is, the more
the value of a
ℓ
is. Consequently, the length of the longitudinal rein-
forcement bars is longer and the total consumption of steel in the
analyzed beam tends to increase. Differently from calculation mo-
del I, considering that the shear force is absorbed by the stirrups
does not influence the value of the offset or the consumption of the
longitudinal reinforcement.
3. Analysis result between calculation
models I and II
The main difference between the model calculations I and II propo-
sed by the Brazilian Standard Code ABNT NBR 6118:2007 is the
consideration of the angle θ as constant and equal to 45° in model I,
and ranging from 30º to 45º in Model II. For both models, the stirrups
can have inclination α ranging from 45° to 90°. For this study, the an-
gle α is considered 90°, since this value is the most commonly used
in structures because of the constructive ease. Another reason for
using vertical stirrups is the inefficiency of inclined stirrups when the-
re is load inversion, which occurs in areas subject to earthquakes.
Both models allow the verification of the concrete compression strut,
(2)
cos
sen
fd9,0
s
A
V
ywd
sw
sw
Equations (3) and (4) present the safety criteria of the transverse
reinforcement. In these equations, the value of f
ctd
, which is the
value for calculating the concrete resistance to the direct traction,
is obtained according to the characteristic resistance to compres-
sion (f
ck
) in equation (5).
(3)
sw c
3Rd
sd
VV V V
(4)
d b f6,0 V
w ctd
0c
(5)
3 2
ck
ctd
f
15,0 f
The Brazilian Standard Code ABNT NBR 6118:2007 [1] indicates
that when the longitudinal tensile reinforcement bar is obtained
considering the balance of loads in the normal cross-section to the
axis of the structural element, the effects caused by the diagonal
crack may be replaced, in the calculation, by the offset (a
ℓ
) of force
in the tensile chord diagram. The offset depends on the design va-
lue of shear force (V
Sd
), the portion (V
c
), the effective depth (“d”) of
the structural element and the a angle of inclination of the stirrups.
Although it is not explicit in the Standard, the value of a
ℓ
must be
limited to the value of the effective depth (d) of the element, and it
is a required parameter for determining the final length of the lon-
gitudinal tensile reinforcement bars. It directly influences the con-
sumption of the reinforced beam. One way to decrease the value
of the offset in model I and, therefore, the length of the longitudinal
reinforcement bars is to consider that the shear force is absorbed
by the stirrups, which implies that the value of V
c
equals zero. This
consideration leads to an increase in the area of the transverse
reinforcement. But, as in a reinforced concrete beam the amount
of steel in the longitudinal reinforcement is much higher than the
volume of steel in the transverse reinforcement, the consumption
of steel in the beam decreases.
2.2 Calculation Model II
For determining the area of the transverse reinforcement bars, the
compression struts in calculation model II have variable inclination
relative to the longitudinal axis of the beam, in the range 30° ≤
θ ≤ 45°. This model assumes that the portion V
c
of reduction of
the design shear resistance (V
Rd3
) is variable due to the alternative
schemes to the trusses. In this hypothesis, the V
c
portion depends