Civil Engineering Reference
In-Depth Information
Figure A.1
Elevation of element of composite beam
We first obtain equations deduced from equilibrium, elasticity and com-
patibility, then eliminate
M
,
F
,
V
and
v
L
from them to obtain a differential
equation relating
s
to
x
, and finally solve this equation and insert the
boundary conditions. These are as follows.
(1)
Zero slip at mid-span, from symmetry, so
s
=
0
when
x
=
0
(A.2)
(2)
At the supports,
M
and
F
are zero, so the difference between the
longitudinal strains at the interface is the differential strain,
ε
c
, and
therefore
d
d
s
x
L
2
=−ε
when
x
=
±
(A.3)
c
Equilibrium
Resolve longitudinally for one component:
d
d
F
x
=−
v
(A.4)
L
Take moments:
d
d
M
x
Vv h
1
2
d
d
M
x
Vv h
1
2
c
a
+=
+=
(A.5)
c
L
c
a
L
s
