Simplified prediction of deflections - Concrete Structures: Stresses and Deformations - page 313

coe cient given here apply only when the cross-section and the reinforce-

ment are constant within the span; other values for the coe cient are

suggested later in this subsection for the more common case when A s and
A

′ s vary within the span. When A s and A ′ s are constant, the tensile stress at

bottom bre in a section at the middle of an end span may be approximated

by the average of the values calculated by Equations (9.13) and (9.18):

fi

( ∆ ψ ) cs

A c
A

( ∆ σ ) cs bot = E c y b − ε cs 1 − (9.20)

2

Note that ( ) cs is the value of curvature which would occur in a simply

supported beam (Equation (9.9) or (9.10) ).
The curvature (

∆ ψ

′ s ). In
actual continuous beams, the bottom reinforcement is larger than the top
reinforcement at mid-span, but the reverse is true at the supports. The curva-
ture (

∆ ψ ) cs due to shrinkage depends mainly on ( A s − A

) cs of any span, when released as a simple beam (Fig. 9.3(b) ), will be
positive at mid-span and negative at the supports. This has the e

∆ ψ

ect of
reducing the absolute value of the statically indeterminate connecting
moment, | M |. It can be shown that in the interior span of a continuous beam
of rectangular cross-section (Fig. 9.3(b) ), the statically indeterminate con-
necting moments M

ff

′ s | is constant, with
the heavier steel at the bottom for only the middle half of the span and at the
top for the remainder of the span. It can also be shown that the de

= 0, when the absolute value | A s − A

ection in
this case is half the value for a simple beam (Equation (9.11) ). For a more
general case, accurate calculation of the value of the connecting moment and
the de

fl

fl ection due to shrinkage must account for the values of A s and A ′ s at

various sections of the span.
As approximation, the change of location of the heavier reinforcement
between top and bottom in a common case may be accounted for by the use
of Equation (9.19) with the reduction coe cient 0.5 for an interior span and

0.7 for an end span. This coe ection of a

simple beam of a constant cross-section, based on the reinforcement at mid-
span. The tensile stress at bottom

cient is to be multiplied by the de fl

bre at the same section may be approxi-
mated by Equation (9.13); this implies ignoring the e

fi

ff ect of the statically

indeterminate connecting moment.

9.3.3 Total deflection

The de fl ections due to applied load, including the e ff ects of creep and shrink-

age for states 1 and 2 are (by superposition):

D 1 = D 1 ( t 0 ) + ( ∆ D 1 ) φ + ( ∆ D 1 ) cs (9.21)

D 2 = D 2 ( t 0 ) + ( ∆ D 2 ) φ + ( ∆ D 2 ) cs (9.22)