Civil Engineering Reference
In-Depth Information
A
B
C
u
B
u
F
IGURE
16.38
Determination of DF
in which
K
AB
(
K
BA
) is the rotational stiffness of the beam AB. The value of
K
AB
depends, as we shall see, upon the support conditions at the ends of the beam. Note
that, from Fig. 16.32 a positive
M
BA
decreases
θ
B
.
=
DISTRIBUTION FACTOR
Suppose that in Fig. 16.38 the out of balance moment at the support B in the beam
ABC to be distributed into the spans BA and BC is
M
B
(
M
BC
) at the first
release. Let
M
BA
be the fraction of
M
B
to be distributed into BA and
M
BC
be the
fraction of
M
B
to be distributed into BC. Suppose also that the angle of rotation at B
due to
M
B
is
θ
B
. Then, from Eq. (16.32)
M
BA
=−
M
BA
+
=
K
BA
θ
B
(16.33)
and
M
BC
=−
K
BC
θ
B
(16.34)
but
M
BA
+
M
BC
+
M
BA
=
0
Note that
M
BA
and
M
BC
are fractions of the balancing moment while
M
B
is the out
of balance moment. Substituting in this equation for
M
BA
and
M
BC
from Eqs (16.33)
and (16.34)
θ
B
(
K
BA
+
−
K
BC
)
=−
M
B
so that
M
B
K
BA
+
θ
B
=
(16.35)
K
BC
Substituting in Eqs (16.33) and (16.34) for
θ
B
from Eq. (16.35) we have
K
BA
K
BA
+
K
BC
K
BA
+
M
BA
=
M
BC
=
(
−
M
B
)
(
−
M
B
)
(16.36)
K
BC
K
BC
The terms
K
BA
/
(
K
BA
+
K
BC
) and
K
BC
/
(
K
BA
+
K
BC
) are the
distribution factors
(DFs)
at the support B.
STIFFNESS COEFFICIENTS AND CARRY OVER FACTORS
We shall now derive values of stiffness coefficient (
K
) and carry over factor (COF)
for a number of support and loading conditions. These will be of use in the solution
