Simplified prediction of deflections - Concrete Structures: Stresses and Deformations - page 312

A c d
¯

κ cs = − y c (9.17)

The subscripts 1 and 2 are employed with κ cs to refer to uncracked and fully

cracked states.
Equations (9.15-16) are derived by combining Equations (9.10) and (9.11)
and (9.17) by comparing Equation (9.10) with Equations (3.16) and (7.27);
equations for a cantilever can be derived in a similar way. The curvature
coe

κ cs2 are to be calculated for the 'determinant' section
which is at mid-span for a simple beam and at the

cients κ cs1 and

fi xed end for a cantilever

(see Section 9.3).
In statically indeterminate structures, hyperstatic forces develop which tend
to reduce the de

ection due to shrinkage. Consider as an example the interior
span of a continuous beam of equal spans (Fig. 9.3(b) ). Assume that the
span shown is su

fl

ciently far from the end spans such that the rotations at A
and B are zero. Use the force method (see Section 4.2) to calculate the static-
ally indeterminate co n necting moments. This gives for a beam of constant
cross-section: M E c ¯ (

) cs represents the curvature if the
beam were simply supported. The curvature due to the connecting moments
is of constant value equal to

= − ∆ ψ ) cs ; where ( ∆ ψ

− ( ∆ ψ ) cs . Thus, the statically indeterminate beam

has no curvature and no de ection due to shrinkage and the concrete stress is

uniform tensile of magnitude:

fl

A c
A

( ∆ σ ) cs = − ε cs E c 1 − (9.18)

Note that the stress in this case depends only on the sum of the reinforcement
areas ( A s

′ s ) not on their locations in the cross-section. For a rectangular
section with 1 per cent reinforcement,

+ A

ε cs = − 300 × 10 −6 and χφ = 2, ( ∆ σ ) cs =

0.45 MPa (0.065 ksi) (Fig. 9.3(b) ).
The statically indeterminate reactions and bending moments caused by
uniform shrinkage in continuous beams of constant cross-section having two
to

fi ve equal spans are given in Fig. 10.7. This fi gure, intended for the e ff ect of

temperature, is also usable for the e ff ect of shrinkage, the only di ff erence is

that the multiplier ( gure represents the change

in curvature due to uniform shrinkage of a simple beam (Equation (9.9) or
(9.10) ). The de

∆ ψ ) used for the values of the fi

ection is largest for the end span and its value at the middle
of the span may be expressed as follows:

fl

De fl ection at the centre of a continuous span

= reduction coe cient × de fl ection of a simple beam. (9.19)

The reduction coe cient for an end span is respectively 0.25, 0.40, 0.36 and

0.37 when the number of spans is 2, 3, 4 and 5. The values of the reduction