Civil Engineering Reference
In-Depth Information
by the timber joist is
E
t
I
t
R
M
t
=
(12.1)
where
E
t
is Young's modulus for the timber and
I
t
is the second moment of area of
the timber section about the centroidal axis,
Gz
. Similarly for the steel plates
E
s
I
s
R
M
s
=
(12.2)
in which
I
s
is the combined second moment of area about
Gz
of the two plates. The
total bending moment is then
1
R
(
E
t
I
t
+
M
=
M
t
+
M
s
=
E
s
I
s
)
from which
1
R
=
M
E
t
I
t
+
(12.3)
E
s
I
s
From a comparison of Eqs (12.3) and (9.7) we see that the composite beam behaves
as a homogeneous beam of bending stiffness
EI
where
EI
=
E
t
I
t
+
E
s
I
s
or
E
t
I
t
+
I
s
E
s
E
t
EI
=
(12.4)
The composite beam may therefore be treated wholly as a timber beam having a total
second moment of area
E
s
E
t
I
t
+
I
s
This is equivalent to replacing the steel-reinforcing plates by timber 'plates' each hav-
ing a thickness (
E
s
/
E
t
)
t
as shown in Fig. 12.2(a). Alternatively, the beam may be
transformed into a wholly steel beam by writing Eq. (12.4) as
E
s
E
t
E
s
I
s
EI
=
I
t
+
so that the second moment of area of the equivalent steel beam is
E
t
E
s
I
t
+
I
s
which is equivalent to replacing the timber joist by a steel 'joist' of breadth (
E
t
/
E
s
)
b
(Fig. 12.2(b)). Note that the transformed sections of Fig. 12.2 apply only to the case
of bending about the horizontal axis,
Gz
. Note also that the depth,
d
, of the beam is
unchanged by either transformation.
The direct stress due to bending in the timber joist is obtained using Eq. (9.9), i.e.
M
t
y
I
t
σ
t
=−
(12.5)
