Civil Engineering Reference
In-Depth Information
(
)
i
can be determined if the stress increments in the preceding increments
are known.
In the step-by-step analysis, a complete analysis of the structure is per-
formed for each time interval. Thus, when the analysis is done for the
i
th
interval, the stress increments in the precedin
g
intervals have been previously
determined. In this way, the initial strains (
∆
ε
)
i
are known values which can
be treated as if they were produced by a change in temperature of known
magnitude.
In the analysis of a plane frame by the displacement method, three nodal
displacements are determined at each joint: translations in two orthogonal
directions and a rotation. With the usual assumption that a plane cross-
section remains plane during deformation, the strain and hence the stress at
any
∆
ε
bre in a cross-section of a member can be calculated from the nodal
displacements at its ends.
In the step-by-step procedure, a linear elastic analysis is executed for each
time interval by the conventional displacement method. The cross-section
properties to be used in this analysis are those of a transformed section
composed of the area of concrete plus
fi
α
i
times the area of steel; where
α
i
is a
ratio varying with the interval and for the
i
th interval:
E
s
(
E
ce
)
i
α
i
=
(5.26)
where
E
s
is the respective modulus of elasticity of prestressed or non-
prestressed steel.
In any interval
i
, the three materials are considered as if th
e
y were sub-
jected to a change of temperature, producing the free strain (
∆
ε
)
i
of known
magnitude. The corresponding stress (
)
i
in the three materials are
unknowns to be determined by the analysis for the
i
th interval; the values
(
∆
σ
)
i
represent the stress due to external loading (if any) applied at the middle
of the
i
th interval plus the stress due to the
∆
σ
fi
ctitious change in temperature
mentioned above.
Analysis of stress due to arbitrary temperature distribut
i
on involves the
following steps. First the strain due to temperature ( (
∆
ε
)
i
in our case)
is arti
M
in each section (see
Equations (2.25) and (2.26) ). This is equivalent to the application of a set
of self-equilibrating forces (see Fig. 5.7 and Equations (5.15) and (5.16) ).
The arti
fi
cially restrained by internal forces
∆
N
and
∆
cial restraint is then removed by application of a set of equal and
opposite forces.
An example of analysis by this method and a listing of a computer pro-
gram which performs the analysis can be found in the references mentioned in
Note 3.
fi
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