Basic HTML Version
3
IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 1
L. A. SPAGNOLO JR | E. S. SÁNCHEZ FILHO |
M. S. L. VELASCO
3.1 Upper-bound solution
An upper-bound solution shows that failure must occur for any
compatible plastic deformation if the rate of the external work of
applied forces on the beam equals the rate of internal energy dis-
sipation for all materials: concrete, steel and CFC.
In the upper-bound approach, it is necessary to hypothesize a
displacement field in discontinuities between the rigid regions of
the beam (these regions are rigid bodies), each with a constant
displacement (or rate of displacement). The rate of internal dis-
sipation depends on the selection of the displacement field, and it
is independent of the applied loads.
The basis for this study is summarized below, followed by the pro-
posed relationships for the upper-bound model.
For beams without shear reinforcement or lightly reinforced beams,
the shear strength is determined by the Cracking Sliding Model
(CSM). “
This model is based on the hypothesis that cracks can be
transformed into yield lines, which have lower sliding strength than
yield lines formed in uncracked concrete
”.
For a crack transformation into a yield line, the internal work per unit
length is calculated. The angle variation between the yield line and
the displacement direction
a
is limited by
ϕ
π
a
ϕ
−≤ ≤
, where
ϕ
is the angle of friction of the concrete. This limitation on
a
shows
that crack sliding must be treated as a plane strain problem.
In this upper-bound analysis of the failure mechanisms, the beams
are loaded in four points divided into a series of three rigid blocks.
These blocks are separated by lines of discontinuity (Figure [1]).
The relationships for the shear capacity are obtained from kine-
matic and yield conditions, and form an upper-bound solution of
Plasticity Theory for reloaded concrete beams.
The theoretical approach proposed obeys all hypotheses of the
CSM, and offers some new conditions to evaluate the ultimate load
of reinforced concrete beams strengthened with glued CFC verti-
cal stirrups. The following analysis considers all formulas valid for
T
beams. The external shear reinforcement can be evaluated in an
additive way, assuming that its behavior is similar to the behavior
of steel stirrups (Figure [2]). The total mechanical ratio of the trans-
crete is given by empirical equations. Although several experiments
concerning shear strengthening have been reported in the literature
([4], [5], [6], [7], [8]), few offer clear and conclusive experimental data.
In this work, a more comprehensive shear experimental study
was performed. This study aimed to establish new experimental
and consistent data in order to evaluate the ultimate shear of the
beams strengthened with
U
stirrups of carbon fiber composites.
2. Research significance
The objective of this study was to investigate the effectiveness
behavior of RC
T
beams strengthened to shear with CFC. This
paper presents an experimental program with the followings objec-
tives: 1) to understand the effect of shear mechanism in the tested
beams; 2) to corroborate a theoretical model; and 3) to supply the
literature with detailed test data that can support further research
in this area. The theoretical model proposed herein was assessed
by comparing the calculated and measured response of eight test
beams, and provided accurate numerical tools that can be exploit-
ed to understand and predict this type of strengthening.
3. Analytical model
Traditional standard analyses and design specifications for the
shear strengthening of reinforced concrete beams use the diago-
nal cracking strength
c
V
as an estimate of the concrete contribu-
tion, and, in general, adopt the classical truss model for calculating
shear reinforcements (steel and CFC) to shear capacity.
The concrete contribution to shear resistance is far more varied
because it is the sum of several internal mechanisms of resistance:
shear carried in the compression zone (uncracked zone), aggre-
gate interlock (shear friction between cracks), and dowel action.
The literature furnishes several formulas for calculating the con-
crete contribution, and the most important codes have selected
significantly different approaches for the
c
V
portion.
This study seeks to investigate a suitable upper-bound solution and
evaluate thismodel by comparison with experimental data ([9], [10], [11]).
Figure 2 – CSM for reinforced concrete beams strengthened with glued CFC vertical stirrups