Civil Engineering Reference
In-Depth Information
Figure 7.2
also shows the isostatic compression curves (dotted) for the
various cases. For the three cases
a/h
=0, 0.25 and 0.5, each isostatic curve
can be approximated by an ellipse. In contrast, for the case of
a/h
=2, the
isostatic curve concentrates near the two loading points. The curve for
a/h
=1
lies somewhere in between.
The effect of transverse compression can now be represented by an
effective transverse compression of intensity
p
, acting uniformly throughout
the shear element. The magnitude of the effective transverse compression
p
is related not only to the shear force
V,
but also to the shear span ratio.
Obviously, the larger the shear span ratio, the smaller the effective
transverse compression will be, given the same shear force
V
. Therefore, the
effective transverse compression
p
can be developed as a function of shear
force
V
and the shear span ratio
a
/
h
.
Consider the case of
a
/
h
=0.5 as shown in Figure 7.2(c). The dotted
isostatic curve indicates the boundary of a possible stress path between the
top and bottom loading points. The width of the load path at the mid-height
can be estimated as
h
/
2,
which is the same as the shear span
a
. Thus, an
estimate of the effective transverse compression is
p
=
V
/
ba
or 2
V
/
bh,
where
b
is the width of the beam. For larger
a
/
h, p
should decrease to zero at certain
value of
a
/
h
. It is reasonable to assume that such a value is
a
/
h
=2. Beyond
a
/
h
=2, the shear behaviour would approach that of a slender beam. When
a
/
h
increases from 0.5 to 2,
p
will decrease not only with
V
/
ba,
but should
incorporate a linear function (4/3-2
a
/
3h
) so that
p
=0 when
a
/
h
=2. The
resulting expression for
p
is
The right-hand side of this equation can be expressed in terms of the
nominal shear across the whole section
V/bh
(7.1)
This expression is plotted in
Figure 7.3.
For
a/h<0.5
, the transverse compression is assumed to remain constant
since the effective area remains essentially the same as shown in Figure 7.2(a)
and (b). The expression of
p
=2
V/bh f
or
a/h
<0.5 is also shown in Figure 7.3.
An effective shear stress
v
in the shear element can be defined by the
following formula
(7.2)
v
=
V/bdv
Thus the stress conditions for the shear element are completely defined
by
p
and
v
. To find the shear strength of the beam is to find the maximum
