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579
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 5
R. BARROS | J.S. GIONGO
tained using calculation model II, the second compares the results
of calculation model II to the results of calculation model I.
Due to the variation of the angle θ ranging from 30° to 45° in
calculation model II, there is a reduction in the value of the rein-
forcement area calculated as the angle of inclination of the strut
approaches 30°. It was also observed that, similar to the reduc-
tion of the design resistance, the relation between the areas of
the transverse reinforcement obtained through calculation model
II occurs due to the tangent of the angle θ relative to the compres-
sion strut, and is not influenced by geometry or by action effects
in the structural element.
Arranging equation (8) for calculating A
sw
and considering stirrups
inclined at 90°, one obtains the relation between the reinforcement
areas of calculation model II and the reinforcement areas of the
same model, with the angle of inclination of the strut equal to 45º.
This relation is represented by equation (10) through which the
graph of Figure 2 is obtained.
(10)
tan
A
A
º45 ,MII , sw
MII , sw
The reduction of the reinforcement area is followed by the reduc-
tion of the design shear resistance on the ruin of the compression
strut, (V
Rd2
). That is, when considering angle θ less than 45°, there
is a penalty of the maximum value allowed for the design shear
force, according to equation (7).
The comparison between the values obtained for the transverse
reinforcement areas in calculation models I and II is difficult due
to several parameters involved in the analysis. It is observed that
for a given value of the design shear force (V
sd
), the portion of
the shear force resisted by the transverse reinforcement (V
sw
) ne-
cessarily have distinct values in the two calculation models. This
occurs because of different considerations that each calculation
through the portion V
Rd2
and subsequent calculation of the area of
the transverse reinforcement A
sw
. A decrease in the design value
is possible due to the truss additional mechanisms, previously pre-
sented in this study. This decrease is constant in model I for any
design value, and in calculation model II, the decrease depends on
the value of the design shear force (V
Sd
).
Thus, the first observation made is that in calculation model II, as
the design force increases, V
c1
force decreases, and equals zero
when the design value of shear force V
Sd
is at its maximum, when
it is equal to the resistance capacity of the compression strut, de-
fined by V
Rd2
. This resistance capacity is constant in calculation
model I and variable in calculation model II, decreasing due to an-
gle θ adopted in the latter. It is stated that this decrease occurs
according to a relation that is independent of the geometry and the
structural element design. By dividing equation (7) by equation (1),
the relation between the design shear resistances V
Rd2
is obtained.
It results in equation (9), which exclusively depends on the angle
of inclination of the compression strut.
(9)
2 sen
V
V
MI ,2Rd
MII ,2Rd
Figure 1 shows the relation obtained through equation (9). Accor-
ding to Figure 1, by adopting angle θ equal to 45° in calculation
model II, one obtains the same value for the V
Rd2
force when using
calculation model I. However, as the inclination angle θ decreases,
the value of V
Rd2
force also decreases. Therefore, the resistance
capacity of the strut decreases to 87% of V
Rd2
value obtained throu-
gh calculation model I when the minimum inclination θ equals 30°.
Several practical examples presented in Barros & Giongo [6] con-
firm these results.
Regarding the calculation of the transverse reinforcement area,
two distinct comparisons are made. The first relates the results ob-
Figure 1 � Relation �et�een design s�ear
resistance relative to compression
strut V , for design models I
Rd2
and II of ABNT NBR 6118:2007 [1]
V
Rd2,MII
/ V
Rd2,MI
80%
85%
90%
95%
100%
45º
42º
39º
36º
33º
30º
Strut angle variation
Figure � � �elation bet�een
transverse reinforcement areas
obtained from design model II for
several angles and considering
θ
=45
A
sw,MII
/ A
sw,MII,
=45º
65%
62%
60%
70% 73%
75%
81%
84%
100%
93%
90%
58%
78%
87%
97%
67%
50%
60%
70%
80%
90%
100%
45º
42º
39º
36º
33º
30º
Strut angle variation