Civil Engineering Reference
In-Depth Information
of a variety of problems. For this purpose we use the slope-deflection equations,
Eqs (16.28) and (16.30). Thus for a span AB of a beam
2
θ
A
+
θ
B
+
L
(
v
A
−
v
B
)
2
EI
L
3
M
AB
=−
and
2
θ
B
+
v
B
)
2
EI
L
3
L
(
v
A
−
M
BA
=−
θ
A
+
In some problems we shall be interested in the displacement of one end of a beam
span relative to the other, i.e. the effect of a sinking support. Thus for, say
v
A
=
0 and
v
B
= δ
(the final two load cases in Table 16.6) the above equations become
2
θ
A
+
2
EI
L
3
L
δ
M
AB
=−
θ
B
−
(16.37)
and
2
θ
B
+
2
EI
L
3
L
δ
M
BA
=−
θ
A
−
(16.38)
Rearranging Eqs (16.37) and (16.38) we have
3
L
δ =−
L
2
EI
M
AB
2
θ
A
+
θ
B
−
(16.39)
and
3
L
δ =−
L
2
EI
M
BA
2
θ
B
+
θ
A
−
(16.40)
Equations (16.39) and (16.40) may be expressed in terms of various combinations of
θ
A
,
θ
B
and
δ
. Thus subtracting Eq. (16.39) from Eq. (16.40) and rearranging we obtain
L
2
EI
(
M
BA
−
θ
B
−
θ
A
=−
M
AB
)
(16.41)
Multiplying Eq. (16.39) by 2 and subtracting from Eq. (16.40) gives
L
−
L
6
EI
(
M
BA
−
θ
A
=−
2
M
AB
)
(16.42)
Now eliminating
θ
A
between Eqs (16.39) and (16.40) we have
L
=−
L
6
EI
(2
M
BA
−
θ
B
−
M
AB
)
(16.43)
We shall now use Eqs (16.41)-(16.43) to determine stiffness coefficients and COFs for
a variety of support and loading conditions at A and B.
Case 1: A fixed, B simply supported, moment
M
BA
applied at B
This is the situation arising when a beam has been released at a support (B) and we
require the stiffness coefficient of the span BA so that we can determine the DF; we
also require the fraction of themoment,
M
BA
, which is carried over to the support at A.
