Civil Engineering Reference
In-Depth Information
M
1
X
2
X
1
F
IGURE
16.23
Indeterminacy
of a portal frame
(a)
(b)
the number of compatibility equations requiring solution becomes too large for hand
computation.
Consider the portal frame shown in Fig. 16.23(a). From Section 16.2 we see that the
frame, together with its foundation, form a single two-dimensional ring and is there-
fore three times statically indeterminate. Therefore we require 3 releases to obtain
the statically determinate primary structure. These may be obtained by removing the
foundation at the foot of one of the vertical legs as shown in Fig. 16.23(b); we then
have two releases of force and one of moment and the primary structure is, in effect,
a cranked cantilever. In this example there would be three compatibility equations
requiring solution, two of translation and one of rotation. Clearly, for a plane, two-bay
portal frame we would have six compatibility equations so that the solution would
then become laborious; further additions to the frame would make a hand method
of solution impracticable. Furthermore, as we shall see in Section 16.10, the moment
distribution method produces a rapid solution for frames although it should be noted
that using this method requires that the sway of the frame, that is its lateral move-
ment, is considered separately whereas, in the flexibility method, sway is automatically
included.
E
XAMPLE
16.12
Determine the distribution of bending moment in the frame
ABCDshown inFig. 16.24(a); the flexural rigidity of all themembers of the frame is
EI
.
4 kN/m
4 kN/m
B
C
C
B
C
B
10 kN
10 kN
x
2
EI
x
3
3m
EI
x
1
EI
1
A
D
A
D
A
D
R
2
1
F
IGURE
16.24
Portal frame
of Ex. 16.12
3.5 m
R
1
(a)
(b)
(c)
Comparison with Fig. 16.23(a) shows that the frame has a degree of statical inde-
terminacy equal to 2 since the vertical leg CD is pinned to the foundation at D. We
therefore require just 2 releases of reaction, as shown in Fig. 16.24(b), to obtain the
statically determinate primary structure. For frames of this type it is usual to neglect
