Civil Engineering Reference
In-Depth Information
A
B
A
B
A
B
Tie
X
1
X
1
11
x
F
IGURE
16.31
Solution for a tied
two-pinned arch
L
L
(a)
(b)
(c)
contributions from the bending of the arch and the axial force in the tie. Thus, with
the usual notation
L
M
0
M
1
EI
F
0
F
1
L
AE
B,H
=
d
s
+
d
x
Profile
0
and
L
M
1
F
1
L
a
11
=
EI
d
s
+
AE
d
x
Profile
0
The compatibility condition is then
B,H
+
a
11
X
1
=
0
SEGMENTAL ARCHES
A segmental arch is one comprising segments having different curvatures or different
equations describing their profiles. The analysis of such arches is best carried out using
a computer-based approach such as the stiffness method in which the stiffness of an
individual segment may be found by determining the force-displacement relationships
using an energy approach. Such considerations are, however, outside the scope of
this topic.
16.9 S
LOPE-
D
EFLECTION
M
ETHOD
An essential part of the computer-based stiffness method of analysis and also of the
moment distributionmethod are the slope-deflection relationships for beamelements.
In these, the shear forces and moments at the ends of a beam element are related to
the end displacements and rotations. In addition these relationships provide a method
of solution for the determination of end moments in statically indeterminate beams
and frames; this method is known as the
slope-deflection method
.
Consider the beam, AB, shown in Fig. 16.32. The beam has flexural rigidity
EI
and
is subjected to moments,
M
AB
and
M
BA
, and shear forces,
S
AB
and
S
BA
, at its ends.
The shear forces and moments produce displacements
v
A
and
v
B
and rotations
θ
A
and
θ
B
as shown. Here we are concerned with moments at the ends of a beam. The usual
sagging/hogging sign convention is therefore insufficient to describe these moments
since a clockwise moment at the left-hand end of a beam coupled with an anticlockwise
