Civil Engineering Reference
In-Depth Information
transformed section. Superposition of the forces in Fig. 5.12(a) and (c)
gives the member-end forces caused by creep (Fig. 5.12(d) ). Following
the notations used with the displacement method in Section 5.2, the
forces in Fig. 5.12(a), (b) and (c) represent respectively {
A
r
}, {
−
F
} and
[
A
u
] {
D
}.
The bending moment at end B of member BC
=
−
1.821
−
0.289
=
−
2.110 MN-m, which is the sum of the bending moment at time
t
0
(see
Table 5.1) and the change due to creep. The bending moments at vari-
ous sections are calculated in a similar way and plotted in Fig. 5.12(e).
The stress distribution at time
t
1
is determined in Table 5.3 by
superposition of:
(a)
Stress at time
t
0
, calculated for
N
=
3.624 MN-m applied on the transformed section at
t
0
. The corres-
ponding strain distribution is de
=
−
0.2431 MN and
M
fi
ned by:
ε
O
(
t
0
)
=
64.9 × 10
−6
and
ψ
280.5 × 10
−6
m
−1
(Table 5.1). The stress values are calculated
by multiplication of the strain by
E
s
(
t
0
)
=
=
200 GPa for the steel or
E
c
(
t
0
)
30 GPa for concrete.
(b) Stress req
u
ired to restrain creep, which is equal to the product of
[
=
−
φ
(
t
1
,
t
0
)
E
c
(
t
1
,
t
0
)/
E
c
(
t
0
)] and the stress in concrete calculated in (a).
(c)
Stress due to
0.9415 MN-m applied
on the age-adjusted transformed section. The corresponding strain
distribution is de
−∆
N
=
−
2.015 MN and
−∆
M
=
fi
ned by:
∆
ε
O
=
−
96.0 × 10
−6
and
∆
ψ
=
92.0 ×
10
−6
m
−1
(Table 5.2).
(d) Stress due to the statically indeterminate forces produced by creep:
axial force
=
−
0.038 MN and moment
=
−
0.289 MN-m applied on
the age-adjusted transformed section.
The stress values for the above four stages and their superposition are
listed in Table 5.3 at the top and bottom
fi
bres of concrete and steel.
The de
fl
ection at G is calculated by superposition of:
(a)
The de
fl
ection at time
t
0
, calculated from the curvature
ψ
(
t
0
), using
Equation (C.8), which gives:
D
(
t
0
)
=
28.46 × 10
−3
m (Table 5.1).
(b) The de
ection due to creep in the released system, calculated from
the curvatures
fl
∆
ψ
giving:
∆
D
=
9.59 × 10
−3
m (Table 5.2).
(c)
The de
fl
ection due to a statically indeterminate moment due to
creep
=
−
0.289 MN-m constant over BC. This gives a de
fl
ection of
−
3.84 × 10
−3
m. Hence, the total de
fl
ection at time
t
1
is
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