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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
hollow section, and h
e
is the measurement of the thickness of the
simulated wall.
(12)
2 sen hA f) 250 / f 1( 50,0 T T
e
e
cd
ck
2Rd
sd
In addition to the verification of the compression strut, it is neces-
sary to perform the verification of the transverse and longitudinal
reinforcement resistance. In the case of transverse reinforcement,
when they present angle a equal to 90°, the resistance condition
is guaranteed by equation (13). In this equation, the value of A
90
represents the cross-sectional area of the number of branches of a
stirrup, which must be in the region of the simulated wall. In relation
to the longitudinal reinforcement, equation (14) must be used.
(13)
g cot
A2 f
s
A
T T
e
ywd
90
3,Rd
sd
(14)
tg A2 f
u
A
T T
e
ywd
e
s
4,Rd
sd
In equation (14), the term u
e
represents the perimeter of the area A
e
.
The torsion longitudinal reinforcement A
sℓ
may have the bars distributed
along the perimeter or concentrated in the corners of the polygon that
defines area A
e
, in a necessarily constant ratio between the portion of
the total calculated area A
sℓ
, and the portion of the perimeter of area A
e
.
Figure 5 presents a rectangular section of the reinforced concrete, in
which the fictitious thickness according to thickness h
e
is delimited. Sin-
ce b
w
is the width and h is the height of the section, the values of A
e
and
u
e
and can be obtained through equations (15) and (16), respectively.
(15)
e
e
w e
h h h b A
(16)
e
e
w
e
hh2 h b2 u
4.1 Torsion effects acting apart
The possibility to vary the angle of inclination of the compression
strut upon the occurrence of torsion in reinforced concrete beam
elements enables an analysis similar to the one performed on item
3 in relation to shear force.
Regarding the analysis of the compression strut, the torsion resistan-
ce design T
Rd2
presents a similar variation of the design shear resis-
tance V
Rd2
. Taking the torsion resistance design with a 45° angle as
reference, there is a reduction of the strength capacity of the strut as
the angle θ approaches 30° according to equation (17). This result is
consistent with the result obtained for the shear force, indicating that
the relation between the strength capacity of the struts depends solely
on the value of the inclination angle θ considered in the analyzes. The
percentage ratio obtained through equation (17) is in Figure 6.
(17)
2 sen
T
T
º45 ,2Rd
2Rd
Figure � � Area A in torsion design model
e
h
e 2
h
b
w
h
e 2
h
e
h
e
Figure � � Relation �et�een resistance
torsion moment relative to compression
strut TRd2, for several strut angles
T
Rd2
/ T
Rd2,
=45º
80%
85%
90%
95%
100%
45º
42º
39º
36º
33º
30º
Strut angle variation