Time-dependent internal forces in uncracked structures: analysis by the force method - Concrete Structures: Stresses and Deformations - page 109

is in the modulus of elasticity to be used in the analysis. With shrinkage,
use the age-adjusted elasticity modulus.

E c ( t 0 )

E c ( t , t 0 ) =

1 + χφ ( t , t 0 )

The bending moment diagram for this frame and the reactions are
derived by a conventional elastic analysis, e.g. by use of the general
force method or by moment distribution; 2 the results are given in
Fig. 4.2(b). Note that the shrinkage ε cs is a negative value; after applica-

tion of the multiplier, the ordinates in Fig. 4.2(b) will have reversed
signs.
To calculate the stress in concrete at any

bre, we should use the
values of the internal forces as calculated by this analysis and
the section properties, A c and I c of the concrete, excluding the
reinforcement.

fi

Example 4.2 Continuous beam constructed in two stages
The continuous prestressed beam ABC (Fig. 4.3(a) ) is cast in two
stages: AB is cast rst and at age 7 days it is prestressed and its forms

removed; span BC is cast in a second stage and its prestressing and
removal of forms are performed when the ages of AB and BC are 60
and 7 days, respectively. Find the bending moment diagram at time
in

fi

nity due to the self-weight of the beam only using the following
creep and aging coe

fi

cients:

φ ( ∞ , 7) = 2.7 χ ( ∞ , 7) = 0.74 φ (60, 7) = 1.1

φ ( ∞ , 60) = 2.3 χ ( ∞ , 60) = 0.78.

Ratio of elasticity moduli for concrete at ages 60 and 7 days are:

E c (60)/ E c (7) = 1.26.

Let t be the time measured from day of casting of AB. A statically
determinate released structure and a system of one coordinate are
shown in Fig. 4.3(b). At t

60, uniform load q is applied on span BC of
the continuous beam ABC, which has moduli of elasticity E c (60) for AB

=