Hall A.S., Archer F.E., Gilbert R.I. Engineering Statics
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For each of the structures shown in Figures
8.3
and
8.5,
there are four reaction
components and one internal hinge. We have seen that these four reactions can be
evaluated using the three equations of equilibrium and
a
fourth equation resulting from
the knowledge that the moment at the hinge must be zero.
From the point of view of calculation,
a
three-hinged arch
is
the same
as
the portal of
Figure
8.5.
Arches are usually curved but may be polygonal like that
of
Figure
8.62~.
A
three-pinned portal (or arch) may have the bases (supports) at different levels,
The
analysis of such problems is essentially the same
as
in the previous examples.
In Example
8.3,
simultaneous equations can be avoided by resolving reactions at
A
onents,
as
in Figure 8.7c, so that
l$
and
R*
act
are now easily found from external equilibrium.
equation of moments to find
4
(or
U,)
now involves an oblique lever arm. For this
reason, it may be easier to use the solution given above.
We have seen that a beam with four reaction components can be solved if it contains an
internal hinge. More generally it can be solved if the
I3.M.
at any point is known. The
hinge
is
a special case when the
I3.M.
is
known
to be zero.
re
8.9a
shows
a
beam continuous over three
spans
and containin
and
F.
The part
EF
is referred to as
a
s~~e~~e~~~~
in
b
now have five reactio~ components
and
two
hinges. There are
th
equili~riu~ and
two
e~uations expressin the fact that
M
==:
O
at
are so~eti~es called
e~~~t~ons
of
con~~tio~).
These five equations are sufficient to
~eter~ine the reactions.
ed
by
the laws of statics if it contains
n
-
3
intern
There
is
a
hinge at
D.
Find the four reaction co~~onent~.
20
kN
30
kN
The circular
bar
of
Find the reactions
a
10
1cN
the
re~ctio~s of
the
str~c~~re
shown
in
Fi
A A
he
in
G
kN
24
kN
10
m
l3
The two-span beam of Figure P8.6 has a hinge at
D.
Find the vertical
reactions at
A,
B
and
C
for the loading shown and draw the S.F. and
B.M.
diagrams.
8
kN
Find the reactions at
A
and
E
for each of the frames shown ili Figure P8.7 and
draw the axial force, shear force and bending moment diagrams.
C
9
kN
t-2
4m
?
(b)
12
kN
G
kN
T
'i'
5
kN
T
4
m
Find the reactions of the three-hinged semi-circular arch of Figure P8.8.
15
kN
E
(i) Find the reactions of the three-hinged semi-circular arch of Figure
P8.9.
(ii) Find the resultant force transmitted through the pin at
D.
t-10
m-~10
~~
Fipre
P8.9
For the portal frame shown in Figure
P8.10,
find the reactions at
A
and
G
and draw the A.F., S.F. and
B.M.
diagrams.
~3
m~
10
kN
C
2
kNlm
D
E
B
The portal frame
of
Figure
P8.11
is pinned at A and
E
and has a roller
support at
D.
For the loading shown find the five reactions, given that
MB
=
-
25
kNm
and
MD
=I:
-20
kNm
(i.e. there is tension on the outside
of the frame at
B
and
D).
45
kN