Civil Engineering Reference
In-Depth Information
Appendix A
Partial-interaction theory
A.1
Theory for simply-supported beam
This subject is introduced in Section 2.6, which gives the assumptions and
notation used in the theory that follows. On first reading, it may be found
helpful to rewrite the algebraic work in a form applicable to a beam with
the very simple cross-section shown in Fig. 2.2. This can be done by
making these substitutions.
Replace
A
c
and
A
a
by
bh
, and
d
c
by
h
.
Replace
I
c
and
I
a
by
bh
3
/12.
Put
k
c
=
n
=
1, so that
E
c
,
E
c
and
E
s
are replaced by
E
.
The beam to be analysed is shown in Fig. 2.15, and Fig. A.1 shows in
elevation a short element of the beam, of length d
x
, distant
x
from the
mid-span cross-section. For clarity, the two components are shown separ-
ated, and displacements are much exaggerated. The slip is
s
at cross-
section
x
, and increases over the length of the element to
s
(d
s
/d
x
) d
x
,
which is written as
s
+
. This notation is used in Fig. A.1 for increments in
the other variables,
M
c
,
M
a
,
F
,
V
c
and
V
a
, which are respectively the
bending moments, axial force and vertical shears acting on the two com-
ponents of the beam, the subscripts c and a indicating concrete and steel.
It follows from longitudinal equilibrium that the forces
F
in steel and
concrete are equal. The interface vertical force
r
per unit length is un-
known, so it cannot be assumed that
V
c
equals
V
a
.
If the interface longitudinal shear is
v
L
per unit length, the force on
each component is
v
L
d
x
. It must be in the direction shown, to be consistent
with the sign of the slip,
s
. The load-slip relationship is
+
pv
L
=
ks
(A.1)
since the load per connector is
pv
L
.
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