Special cases of uncracked sections and calculation of displacements - Concrete Structures: Stresses and Deformations - page 74

The value ∆ σ ps which is

generally not known. Thus for calculation of t he total loss due to creep,
shrinkage and relaxation, an assumed value of

χ r depends upon the magnitude of the total loss

∆ σ pr is substituted in Equation

(3.4) to give a P c . This answer is used to obtain an improved

reduced relaxation value and Equation (3.4) is used again to calculate a better
estimate of

fi rst estimate of ∆

∆ P c . In most cases, a fi rst estimate of χ r = 0.7 followed by one

iteration gives su cient accuracy.

3.2.1 Changes in strain, in curvature and in stress due to
creep, shrinkage and relaxation
The changes in axial strain at O or in curvature during the period ( t - t 0 ) may
be expressed as the sum of the free shrinkage, the creep due to the prestress
and external applied loads plus the instantaneous strain (or curvature) plus
creep produced by the force ∆ P c which acts at a distance y st below O. Thus,

P c
E c ( t , t 0 ) A c

∆

∆ ε O = ε cs ( t , t 0 ) + φ ( t , t 0 ) ε O ( t 0 ) + (3.10)

P c y st
E c ( t , t 0 ) I c

∆

.

∆ ψ = φ ( t , t 0 ) ψ ( t 0 ) + (3.11)

The change in concrete stress at any fi bre due to creep, shrinkage and

relaxation is

∆ σ c = ∆

A c + ∆

P c P c y st

I c

y

where y is the coordinate of the bre considered; y is measured downwards

from the centroid of concrete area. Substitution of I c

fi

= A c r c in the last

equation gives

∆ σ c = ∆

P c
A c

yy st
r c

1 + (3.12)

The changes in stress in the prestressed steel and in the non-prestressed
reinforcement caused by creep, shrinkage and relaxation are:

∆ σ ns = E st ( ∆ ε O + y ns ∆ ψ ) (3.13)

∆ σ ps = E st ( ∆ ε O + y ps ∆ ψ ) + ∆ σ pr (3.14)

where y ns and y ps are the y coordinates of a non-prestressed and prestressed
steel layer, respectively.