Civil Engineering Reference
In-Depth Information
8.4.3
Experimental evidence
There exists a wealth of published shear test results, which have been
compared with the plastic solution, albeit exclusively with the flexural
capacity prediction in the Òshear strengthÓ formulation, Eqn (8.11). Nielsen and
Braestrup (1978) reported a series of five rectangular, simply supported,
prestressed beams under two-point loading. The beam parameters were: depth
h: 360 mm, concrete cylinder strength
f
c
:55 N/mm
2
, degree of reinforcement
f
:0.21 (including both bottom and top strands), and shear span ratio
a/h:
0.5,
1.0, 2.0, 3.0 and 4.0. The latter beam failed in flexure, whereas shear failure
was obtained for the four beams with lower shear span ratios.
The ultimate loads of all five beams were in excellent agreement with
Eqn (8.11), with an effectiveness factor
v
=0.46, thus the beams were close
to being over-reinforced. Comparison with a number of over-reinforced
beams (
v
/2) from the literature showed some scatter around the prediction
corresponding to v=0.6.
It appears that in comparing with test results, as well as in practical
applications of the solution, the crux of the matter is the assignment of a
value to the effectiveness factor. As mentioned in Section 8.3.1 the reduced
effective concrete strength reflects the limited ductility of concrete, which
depends primarily on the strength level
f
c
. In addition, however, the
effectiveness factor must account for other neglected features, notably the
size effect, the tensile concrete strength, and the state of stress at failure.
The amount of stress redistribution increases with the flatness of the
compressive concrete strut, wherefore the effectiveness factor is expected to
be a decreasing function of the shear span ratio
a/h
. On the other hand, the
neglect of the tensile concrete strength leads to an underestimation of the rate
of internal work in the yield line
(
Figure 8.10
), which is greater for flatter yield
lines, where the relative displacement rate is closer to the yield line normal.
Consequently, the tensile strength leads to an increased effectiveness factor for
higher shear span ratios, cancelling out the above effect.
The development of cracking that eventually leads to failure is basically a
fracture mechanics phenomenon, which is scale dependent. The
effectiveness factor is therefore a decreasing function of the absolute
dimensions of the beam, e.g. represented by the depth
h
.
Finally, experience shows a beneficial influence of the reinforcement,
possibly due to dowel action, in addition to the dependence upon the
reinforcement degree
f‡
. Hence the effectiveness factor is also an increasing
function of the geometrical reinforcement ratio
F
=
A
s
/A
c
.
A comprehensive investigation of published test results has been carried
out by G.W.Chen (1988). The conclusion is that the effectiveness factor for
rectangular, non-prestressed beams can be expressed by the formula:
r
(8.25)
