Civil Engineering Reference
In-Depth Information
The vertical shear at section
x
is
wx
, so
V
c
+
V
a
=
wx
(A.6)
Now
1
2
(
h
c
+
h
s
)
=
d
c
, so from Equations A.5 and A.6,
d
d
M
x
d
d
M
x
c
a
(A.7)
+
+
wx
=
v d
L
c
Elasticity
In beams with adequate shear connection, the effects of uplift are negligi-
ble in the elastic range. If there is no gap between the two components,
they must have the same curvature,
, and simple beam theory gives the
moment-curvature relations. Using Equation 2.19 for
E
φ
c
, then
M
EI
nM
kEI
a
c
φ
=
=
(A.8)
sa
cac
The longitudinal strains in concrete along AB (Fig. A.1) and in steel
along CD are:
1
2
h
nF
kEA
ε
=
φ
−
−
ε
(A.9)
c
c
AB
cac
1
2
h
F
EA
ε
=−
φ
+
(A.10)
CD
s
aa
where
ε
c
is the free shrinkage strain of the concrete, taken as positive.
Compatibility
The difference between
ε
AB
and
ε
CD
is the slip strain, so from Equations
A.9 and A.10, and putting
2
(
h
c
+
h
s
)
=
d
c
,
d
d
s
x
F
E
⎛
⎜
n
kA A
1
⎞
⎟
−
(A.11)
=−
φ
d
+
ε
c
c
a
c
c
a
It is now possible to derive the differential equation for
s
. Eliminating
M
c
and
M
a
from Equations A.7 and A.8,
kI
n
d
d
φ
⎛
⎝
⎞
⎠
cc
E
+
I
+=
wx
v d
(A.12)
a
a
L
c
x
From Equations A.1 and 2.22,
