Civil Engineering Reference
In-Depth Information
S
y
S
y
S
z
'cut'
A
h
0
q
s
p
s
S
z
S
s,
0
F
IGURE
10.19
Determination of
shear flow value at
the origin for
s
in a
closed section beam
Moment
centre
j
0
(a)
(b)
is effectively a constant shear flow round the section, corresponds to the pure torque
produced by the shear load transference. Clearly different positions of the 'cut' will
result in different values for
q
s
,0
since the corresponding 'open section' beams have
different shear centre positions.
Equating internal and external moments in Fig. 10.19(a), we have
pq
s
d
s
pq
b
d
s
q
s
,0
p
d
s
S
z
η
0
+
S
y
ξ
0
=
=
+
where
denotes integration taken completely round the section. In Fig. 10.19(a) the
elemental area
δ
A
is given by
1
2
p
δ
A
=
δ
s
Thus
p
d
s
2
d
A
=
or
p
d
s
=
2
A
where
A
is the area enclosed by the mid-line of the section wall. Hence
pq
b
d
s
S
x
η
0
+
S
y
ξ
0
=
+
2
Aq
s
,0
(10.25)
If the moment centre coincides with the lines of action of
S
z
and
S
y
then Eq. (10.25)
reduces to
pq
b
d
s
0
=
+
2
Aq
s
,0
(10.26)
The unknown shear flow
q
s
,0
follows from either of Eqs. (10.25) or (10.26). Note that
the signs of the moment contributions of
S
z
and
S
y
on the left-hand side of Eq. (10.25)
depend upon the position of their lines of action relative to the moment centre. The
values given in Eq. (10.25) apply only to Fig. 10.19(a) and could change for different
moment centres and/or differently positioned shear loads.
