Civil Engineering Reference
In-Depth Information
w
A
B
A
B
M
A
EI
EI
R
A
1
R
B
1
R
A
2
R
B
2
L
L
F
IGURE
13.23
Alternative solution
of Ex. 13.19
(a)
(b)
F
IGURE
13.24
Practical examples
of fixed beams
(a)
(b)
A
B
M
A
M
B
F
IGURE
13.25
Support reactions in
a fixed beam
R
A
R
B
BUILT-IN OR FIXED-END BEAMS
In practice single-span beams may not be free to rotate about their supports but are
connected to them in a manner that prevents rotation. Thus a reinforced concrete
beam may be cast integrally with its supports as shown in Fig. 13.24(a) or a steel beam
may be bolted at its ends to steel columns (Fig. 13.24(b)). Clearly neither of the beams
of Fig. 13.24(a) or (b) can be regarded as simply supported.
Consider the fixed beam of Fig. 13.25. Any system of vertical loads induces reactions
of force and moment, the latter arising from the constraint against rotation provided
by the supports. There are then four unknown reactions and only two possible equa-
tions of statical equilibrium; the beam is therefore statically indeterminate and has
two redundancies. A solution is obtained by considering known values of slope and
deflection at particular beam sections.
E
XAMPLE
13.19
Figure 13.26(a) shows a fixed beam carrying a central concen-
trated load,
W
. Determine the value of the fixed-end moments,
M
A
and
M
B
.
Since the ends A and B of the beam are prevented from rotating, moments
M
A
and
M
B
are induced in the supports; these are termed fixed-end moments. From symmetry
we see that
M
A
=
M
B
and
R
A
=
R
B
=
W
/
2.
