Civil Engineering Reference
In-Depth Information
2.2.1
No shear connection
It is assumed first that there is no shear connection or friction on the
interface AB. The upper beam cannot deflect more than the lower one,
so each carries load
w
/2 per unit length as if it were an isolated beam of
second moment of area
bh
3
/12, and the vertical compressive stress across
the interface is
w
/2
b
. The mid-span bending moment in each beam is
wL
2
/16. By elementary beam theory, the stress distribution at mid-span is
given by the dashed line in Fig. 2.2(c), and the maximum bending stress
in each component,
σ
, is given by
My
I
wL
2
12
h L
bh
3
8
2
max
σ
=
=
=
(2.1)
16
bh
3
2
2
The maximum shear stress,
, occurs near a support. The two parabolic
distributions given by simple elastic theory are shown in Fig. 2.2(d); and
at the centre-line of each member,
τ
3
24
wL
13
8
wL
bh
τ
=
=
(2.2)
bh
The maximum deflection,
δ
, is given by the usual formula
δ
(/)
52
384
wL
EI
5
384 2
wL
Ebh
12
5
64
wL
Ebh
4
4
4
=
=
=
(2.3)
3
3
The bending moment in each beam at a section distant
x
from mid-span is
M
x
=
w
(
L
2
−
4
x
2
)/16, so that the longitudinal strain
ε
x
at the bottom fibre
of the upper beam is
My
EI
w
Ebh
Lx
3
max
ε
x
=
=
(
−
4
)
(2.4)
2
2
8
2
There is an equal and opposite strain in the top fibre of the lower beam, so
that the difference between the strains in these adjacent fibres, known as
the
slip strain
, is 2
ε
x
.
It is easy to show by experiment with two or more flexible wooden
laths or rulers that, under load, the end faces of the two-component beam
have the shape shown in Fig. 2.3(a). The slip at the interface,
s
, is zero
at
x
=
0 (from symmetry) and a maximum at
x
=
±
L
/2. The cross-section
at
x
0 is the only one where plane sections remain plane. The slip strain,
defined above, is not the same as slip. In the same way that strain is rate
of change of displacement, slip strain is the rate of change of slip along
the beam. Thus from Equation 2.4,
=
