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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 5
R. BARROS | J.S. GIONGO
inclination of the compression struts. Depending on the calculation
model adopted, there are different values for the final length of the
longitudinal reinforcement beam, due to the offset in the diagram
of loads on the tensile chords, as shown in item 2. As the objective
of this study is to consider only values of transverse reinforcement,
the areas of longitudinal reinforcement bars were not considered.
However, this consideration is important when comparing the total
consumption of reinforcement in reinforced concrete beam.
Thus, the comparison between the areas of the transverse rein-
forcement utilizing calculation models I and II depends on three
parameters previously described: the intensity of the design value
of shear force, angle θ of inclination of the strut and the concre-
te class (which, according to the Brazilian Standard Code ABNT
NBR 6118:2007 [1], can be C25, C30, C35, C40, C45 and C50).
Equation 11 shows the relation between the areas of the transver-
se reinforcement obtained according to calculation models I and
II. In this expression, the portion V
Rd2
refers to calculation model II
and may be obtained through equation (7).
(11)
tan
V V
V
V V
V V
V
A
A
0c
Sd
0c
0c
2Rd
Sd
2Rd
Sd
MI , sw
MII , sw
The results obtained when calculating the area of the transverse
reinforcement using calculation models I and II are presented. In
order to facilitate the analysis of the results, the design value of
shear force (V
Sd
) was defined as a percentage of the value of the
design shear resistance (V
Rd2
) in model II when considering the
angle of inclination of the strut equal to 45º. Four percentages were
used for calculating the design value of shear force: 20%, 40%,
60% and 80% of the value of (V
Rd2
). Thus, it is possible to separate-
ly observe the influence of the design value of shear force intensity
and the concrete class on the process of obtaining the transverse
reinforcement area. It is noted that, as the angle of inclination of
the strut decreases from 45º to 30º, there is a percentage reduction
of the relation between the areas of the reinforcement obtained
through calculation models II and I. According to the results obtai-
ned with calculation model I, the same value for the design shear
resistance (V
Rd2
) is obtained when using calculation model II with
an inclination angle of the strut equal to 45°, but with a larger area
of the transverse reinforcement.
It is also stated that calculation model II always presents higher
reinforcement than that obtained with calculation model I when
using strut angles ranging from 40° to 45°. When using the strut
angle equal to 39°, the reinforcement area value obtained with mo-
del II results in the same obtained when using calculation model I.
However, it presents a lower resistance capacity of the compres-
sion strut. For angle values ranging from 30º to 38º, calculation
model II leads to smaller values of the transverse reinforcement
area, with minimum value when θ equals 30°.
For the same class of concrete, the reduction rate is not affected
by the intensity of the action effects and remains constant for every
design value of shear force (V
Sd
), as shown in Table 1. However, it
is observed that a change in the concrete class causes small mo-
difications in the percentage of reduction. Comparing classes C25
and C50, the relation between the transverse reinforcement areas
obtained through calculation model II and calculation model I with
the strut angle of 45° decreased from 122% to 119%. Similarly,
when considering the strut angle equal to 30° for these classes, the
ratio between the areas ranged from 73% to 71%. The influence
of the concrete class can be observed in the graphs of Figure 3.
4. Linear elements subjected to torsion
The Brazilian Standard Code ABNT NBR 6118:2007 [1] fixes con-
ditions for the verification of reinforced concrete beam elements
subjected to torsion combined with other structural loads, assu-
ming a resistant model (space truss) which is defined based on a
structural element with hollow section equivalent to the structural
element to be calculated. That allows the angle θ of the inclina-
tion strut to have its value ranging from 30º to 45º. But, it does
not have two different calculation models such as the reinforced
concrete beam elements subjected to shear force. The standard
code requires the angles of the inclination strut to be the same for
determining resistance when there is a combination of torsion and
shear force.
Figure 4 shows the resultant forces in the reinforcement bars in a
structural part submitted exclusively to torsion. The R
sℓ
force re-
presents the results of tensile stresses in the longitudinal reinfor-
cement bars distributed along the element section. The R
s90
force
is the resultant of tensile stresses on stirrups positioned at 90º in
relation to the part axis. The R
cw,tor
forces represent the resultant of
the compression loads in the compressed struts.
Based on the results presented by Leonhardt & Mönnig [3] and on
the design of the space truss, the Brazilian standard Code ABNT
NBR 6118:2007 [1] indicates a calculation model for reinforced
concrete beam elements subjected to torsion. It is assumed that
the transverse reinforcement have inclinations ranging from 45º
to 90º, and that the design torsion must be less than or equal to
the resistance capacity of the compression strut (T
Rd2
), which is
calculated through expression (12). In this expression, the value of
A
e
represents the area delimited by the medium line of the wall of
Figure � � Mode� o� �e�tion �i��ed �ith si���e
torsion - Leonhardt & Mönnig [3]
R
R
s90
R
s90
Cracks
Stirrups with 90º
Compression strut - Forces R
cw,tor
s
R
cw,tor
R
s
R
s
R
s