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1. Introduction
ABNT NBR 6118:2007 [1] sets conditions for the verification of
reinforced concrete beam elements subjected to shear force, pos-
sibly combined with other action effects, allowing two calculation
models. Such models are based on the parallel chord truss analo-
gy, initially studied by Mörsch [2], in which the angle θ of inclination
of the strut can be considered constant and equal to 45º (model I)
or ranging from 30° to 45º (model II). The transverse reinforcement
may present inclination between 45º and 90º, and it is usually used
in ties projects with an inclination of 90º.
The aim of this paper is to present the analysis of the percentage
differences obtained from the values of transverse reinforcement
areas for reinforced concrete beams related to calculation models
I and II. The method consists of theoretical analyzes based on
the equations of the calculation models presented in the Brazilian
Standard Code ABNT NBR 6118:2007 [1]. The characteristics of
each calculation model and a comparative study of the main pa-
rameters that compose each model are presented. In this respect,
analyzes of the calculation results of the reinforcement areas deri-
ved from the isolated action of shear force and torsion, as well as
their combined action, are presented.
1.1 Initial studies
The classical calculation model of reinforced concrete beam ele-
ments submitted to shear force is based on the classic truss of
Mörsch [2], which considers the beam behavior analogous to an
isostatic truss, in which the upper and lower chords are parallel
to each other, and represented respectively by the region of the
compression concrete and the longitudinal tensile reinforcement
bars of the beam.
Between the chords there are compression concrete struts incli-
ned at 45º degrees to the longitudinal axis of the beam, and a
tie inclined at an angle α which can vary from 45° to 90°, located
transversely to the concrete cracks.
In the truss model, the loads in the compression strut and tie increase
in intensity from the center of the beam towards the support condi-
tions, where the shear force presents its maximum value. By contrast,
the forces on the compression chord and longitudinal tensile reinfor-
cement bars reach their peak in regions near the middle of the span.
Tests conducted by Leonhardt & Mönnig [3] found that the mea-
sured stresses in the transverse reinforcement were lower than
those expected in the design, implying that the theoretical model
of classical truss led to high values of transverse reinforcement.
The observations made by Leonhardt & Mönnig [3] are due to se-
veral factors. The first is that the compression chord is relatively
inclined when compared to the tensile chord, allowing a direct ab-
sorption of a portion of the shear force on the concrete. Due to this
inclination, the R
st
load acting on the longitudinal reinforcement is
greater than the R
cc
load acting on the compression chord.
Regarding the diagonals, the cracks and the struts between them
are variably inclined in relation to the longitudinal axis of the beam
presenting inclinations lower than 45°.
The relative stress decrease in the transverse reinforcement is
due to alternative schemes of shear force absorption developed
with the truss. These schemes are the Arc effect, the interlocking
aggregate effect and the dowel effect of the longitudinal reinforce-
ment bars. Because of these alternative mechanisms, a V
c
value
reduction of the shear force is considered. Its objective is to appro-
ximate the theoretical model to the actual model.
2. Calculation models for shear force
The use of calculation models I and II presented in the Brazilian
Standard Code ABNT NBR 6118:2007 [1] are widely discussed.
Mota & Laranjeiras [4] concluded via electronic mailing list that cal-
culation model I is not a particular case of calculation model II. Sava-
ris & Garcia [5] developed a study on the optimum angle to the strut
and tie to achieve minimum consumption of the area of the trans-
verse and longitudinal reinforcement bars. The researchers found
that model I leads to a minimum consumption of the reinforcement if
used with stirrups inclined between 55° and 60°. However, the use
of inclined stirrups is not often adopted because it requires greater
care in detail and beams assemble during the construction stages.
Barros & Giongo [6] stated a relation between the areas of the
transverse reinforcement bars, obtained according to calculation
models I and II, and it does not depend on the geometry of the
structural element or the intensity of the actions.
The calculation models presented in the Brazilian Standard Code
ABNT NBR 6118:2007 [1] to verify the safety of reinforced con-
crete beam elements subjected to shear force are similar in some
situations. For both models, it is considered that all elements must
have a minimum transverse reinforcement consisting of stirrups,
which the minimal geometric rate (ρ
sw,mín
) depends on the average
resistance to the tensile concrete and the characteristic resistance
to the steel flow of the transverse reinforcement.
Both models permit stirrups with variable inclination ranging from
45° to 90°, and stipulate a maximum value for the V
Sd
design value
of shear force. This value considers the resistance capacity of the
compression strut, named V
Rd2
, which its expression depends on
each calculation model. This limitation intends to prevent that ele-
ments subjected to shear force are ruined due to a rupture of the
compression strut concrete.
The expressions presented in the next items and presented in the
Brazilian Standard Code ABNT NBR 6118:2007 [1] and follow the
indications of the Model Code CEB-FIP [7]. These deductions can
be found at Mangini [8].
2.1 Calculation model I
Calculation model I allows the compression struts to have constant
θ inclination in relation to the longitudinal axis of the element with
a value of 45º. Furthermore, it considers a V
c
reduction portion of
the design shear resistance of V
Rd3
due to the schemes used as an
alternative to the truss schemes previously described. In cases of
flexure and flexotraction in which the neutral axis crosses the sec-
tion of the structural element, the V
c
portion equals V
c0
and its value
is constant and independent of the design value of shear force, V
sd
.
The concrete compression strut is verified through equation (1), and
the calculation of the transverse reinforcement is given by equation (2).
(1)
d b f) 250 / f 1( 27,0 V V
w cd
ck
2Rd
sd
577
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 5
R. BARROS | J.S. GIONGO